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Skibidi Toilet
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Explain it to me quickly
Skibidi Toilet: what is this bizarre viral YouTube series – and does it deserve the moral panic?
More than 38 million people have subscribed to the series about terrifying animated heads that live in toilets – but fears of Skibidi Toilet Syndrome may be slightly extreme
Alex McKinnon explains it to Steph Harmon
Sun 21 Jan 2024 21.50 EST
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Alex, I read a headline saying Russian Cops Forced to Investigate YouTube’s Famous ‘Skibidi Toilet’ Series. As a series of words, what does that mean?
Brrr Skibidi dop dop dop yes yes, Steph. Skibidi dobidi dib dib.
I think that pretty much explains itself.
Thanks, I hate it. But it has 135m views (!) and I’m going to need more info.
Fine. Skibidi Toilet is an ongoing animated YouTube web series by Georgian content creator Alexey Gerasimov. Since it began in February, his YouTube channel, DaFuq!?Boom!, is now just outside the top 100 YouTube channels in the world by number of subscribers. There are now more than 70 episodes, with new videos uploaded every few days – although the time between new episodes has gotten longer as their length and production quality increases.
Skibidi Toilet’s popularity is mainly being driven by children under the age of 13 – it’s the first meme that members of gen Z are lamenting not understanding because they’re too old. The young age of most Skibidi Toilet fans, combined with the show’s unsettling aesthetics and violence, make it perfect fodder for a new moral panic over how the internet is poisoning children’s brains.
Parenting websites and TikTok influencers, particularly in Indonesia, are already warning about the apparent dangers of “Skibidi Toilet Syndrome” – and the Russian authorities have gotten involved.
TikTok’s nine-month cruise: what is it and and why can’t I stop watching?
Skibidi Toilet Syndrome???? What are the symptoms please?
Look it’s not yet recognised in the DSM-V – it’s more a catch-all term for what parents believe is concerning behaviour their children display after watching the show. Parents have documented their kids becoming “obsessed” with Skibidi Toilet, upset or angry after being restricted or banned from watching it, or sitting in baskets or boxes and acting like the Skibidi toilets, which seems more cute than terrifying but what do I know.
Police in Moscow got wind of it after a father asked authorities to investigate whether the videos “have a detrimental effect on children”. Russian lawmakers have been especially prone to looking for signs of moral decay in internet culture – in 2015, the Duma passed laws forbidding memes that mock or satirise public figures.
What’s the show actually about?
The first few episodes show a city and its inhabitants steadily being taken over by the Skibidi toilets, the terrifying animated heads that live in toilets and continually sing a mashup of the songs Give It to Me by Timbaland and Dom Dom Yes Yes by Biser King. The cameraheads – well-dressed men with CCTV cameras for heads who serve as the series’ protagonists – emerge as an underground resistance movement; it quickly becomes an all-out war between ever more powerful and destructive variants of both sides.
This is odd to admit, but I found myself pulled into the world of Skibidi Toilet. There’s a clear narrative arc developing and there are plot twists, betrayals, humour, killer action scenes and a couple of moments where I probably felt more than I’m supposed to.
You mentioned “a new moral panic over how the internet is poisoning children’s brains”. I feel as though there’s one of these every few months?
Handwringing about the internet turning kids into psychopaths has become a recurring feature of the contemporary news cycle. In the last few years social media has been blamed for making kids and teenagers eat laundry detergent, steal items from their schools and develop symptoms of Tourette’s syndrome.
There’s a political element too. Far-right politicians in the US and elsewhere are fundraising and campaigning off ludicrous claims that TikTok is brainwashing kids into questioning their sexual and gender identities, supporting Hamas or becoming bait for child sex traffickers.
None of this is new. Millennials will remember the pearl-clutching around weird internet ephemera such as Slender Man and bigger cultural markers including the Grand Theft Auto video game series. This isn’t to say there isn’t online content that children should be protected from, such as the disturbing bootleg YouTube videos of Peppa Pig being mauled at the dentist. But if every piece of content that kicked off a moral panic merited one, the internet-using kids of the world would long since have become the Children of the Corn.
So the verdict on Skibidi Toilet?
It’s kind of fun! There are definitely parts that a small child might find frightening (a lot of the episodes end with a Skibidi toilet lunging at the camera) but once you get past the red-eyed toilet monsters it’s a lot less creepy than much of the discourse around it has made it out to be.
In any case, a lot of the best kids’ entertainment is weird and dark. Roald Dahl’s books are full of murderous school principals and kids getting mutilated in comedic ways. Hansel and Gretel had to push a witch into the oven! If you’re worried about your kid suddenly acting like a singing toilet, watch some Skibidi Toilet with them – it might even become a guilty pleasure.
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List of trigonometric identities
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Edit
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Pythagorean identities
edit
Main article: Pythagorean trigonometric identity
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity
1
+
cot
2
csc
2
�{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }, and the red triangle shows that
tan
2
�
+
sec
2
�{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }.
The basic relationship between the sine and cosine is given by the Pythagorean identity:
sin
2
�
+
cos
2
1
,
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
where
sin
2
�{\displaystyle \sin ^{2}\theta } means
(
sin
�
)
2
{\displaystyle (\sin \theta )^{2}} and
cos
2
�{\displaystyle \cos ^{2}\theta } means
(
cos
�
)
2
.
{\displaystyle (\cos \theta )^{2}.}
This can be viewed as a version of the Pythagorean theorem, and follows from the equation
�
2
+
�
1
{\displaystyle x^{2}+y^{2}=1} for the unit circle. This equation can be solved for either the sine or the cosine:
sin
±
1
−
cos
2
�
,
cos
±
1
−
sin
2
�
.
{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}
where the sign depends on the quadrant of
�
.
{\displaystyle \theta .}
Dividing this identity by
sin
2
�{\displaystyle \sin ^{2}\theta },
cos
2
�{\displaystyle \cos ^{2}\theta }, or both yields the following identities:
1
+
cot
2
csc
2
�
1
+
tan
2
sec
2
�
sec
2
�
+
csc
2
sec
2
�
csc
2
�
{\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \&1+\tan ^{2}\theta =\sec ^{2}\theta \&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}
Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):
Each trigonometric function in terms of each of the other five.[1]
in terms of
sin
�{\displaystyle \sin \theta }
csc
�{\displaystyle \csc \theta }
cos
�
sec
�
tan
�
cot
�
sin
{\displaystyle \sin \theta =}
sin
�{\displaystyle \sin \theta }
1
csc
�{\displaystyle {\frac {1}{\csc \theta }}}
±
1
−
cos
2
�
±
sec
2
�
−
1
sec
�
±
tan
�
1
+
tan
2
�
±
1
1
+
cot
2
�
csc
{\displaystyle \csc \theta =}
1
sin
�{\displaystyle {\frac {1}{\sin \theta }}}
csc
�{\displaystyle \csc \theta }
±
1
1
−
cos
2
�
±
sec
�
sec
2
�
−
1
±
1
+
tan
2
�
tan
�
±
1
+
cot
2
�
cos
{\displaystyle \cos \theta =}
±
1
−
sin
2
�{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}
±
csc
2
�
−
1
csc
�{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}
cos
�
1
sec
�
±
1
1
+
tan
2
�
±
cot
�
1
+
cot
2
�
sec
{\displaystyle \sec \theta =}
±
1
1
−
sin
2
�{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}
±
csc
�
csc
2
�
−
1
{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}
1
cos
�
sec
�
±
1
+
tan
2
�
±
1
+
cot
2
�
cot
�
tan
{\displaystyle \tan \theta =}
±
sin
�
1
−
sin
2
�{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}
±
1
csc
2
�
−
1
{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}
±
1
−
cos
2
�
cos
�
±
sec
2
�
−
1
tan
�
1
cot
�
cot
{\displaystyle \cot \theta =}
±
1
−
sin
2
�
sin
�{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}
±
csc
2
�
−
1
{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}
±
cos
�
1
−
cos
2
�
±
1
sec
2
�
−
1
1
tan
�
cot
�
Reflections, shifts, and periodicity
edit
By examining the unit circle, one can establish the following properties of the trigonometric functions.
Reflections
edit
Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 – theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 – theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 – theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 – theta) transforms the coordinates to (a,-b).
Transformation of coordinates (a,b) when shifting the reflection angle
�{\displaystyle \alpha } in increments of
�
4
{\displaystyle {\frac {\pi }{4}}}.
When the direction of a Euclidean vector is represented by an angle
�
,
{\displaystyle \theta ,} this is the angle determined by the free vector (starting at the origin) and the positive
�
{\displaystyle x}-unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive
�
{\displaystyle x}-axis. If a line (vector) with direction
�{\displaystyle \theta } is reflected about a line with direction
�
,
{\displaystyle \alpha ,} then the direction angle
�
′{\displaystyle \theta ^{\prime }} of this reflected line (vector) has the value
�
′2
�
−
�
.
{\displaystyle \theta ^{\prime }=2\alpha -\theta .}
The values of the trigonometric functions of these angles
�
,
�
′{\displaystyle \theta ,\;\theta ^{\prime }} for specific angles
�{\displaystyle \alpha } satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.[2]
�{\displaystyle \theta } reflected in
�0
{\displaystyle \alpha =0}[3]
odd/even identities
�{\displaystyle \theta } reflected in
�
4
{\displaystyle \alpha ={\frac {\pi }{4}}}
� reflected in
�
2
� reflected in
�3
�
4
� reflected in
��
compare to
0
sin
(
−
�
−
sin
�{\displaystyle \sin(-\theta )=-\sin \theta }
sin
(
�
2
−
�
cos
�{\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }
sin
(
�
−
�
+
sin
�
sin
(
3
�
2
−
�
−
cos
�
sin
(
2
�
−
�
−
sin
(
�
sin
(
−
�
)
cos
(
−
�
+
cos
�{\displaystyle \cos(-\theta )=+\cos \theta }
cos
(
�
2
−
�
sin
�{\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }
cos
(
�
−
�
−
cos
�
cos
(
3
�
2
−
�
−
sin
�
cos
(
2
�
−
�
+
cos
(
�
cos
(
−
�
)
tan
(
−
�
−
tan
�{\displaystyle \tan(-\theta )=-\tan \theta }
tan
(
�
2
−
�
cot
�{\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }
tan
(
�
−
�
−
tan
�
tan
(
3
�
2
−
�
+
cot
�
tan
(
2
�
−
�
−
tan
(
�
tan
(
−
�
)
csc
(
−
�
−
csc
�{\displaystyle \csc(-\theta )=-\csc \theta }
csc
(
�
2
−
�
sec
�{\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }
csc
(
�
−
�
+
csc
�
csc
(
3
�
2
−
�
−
sec
�
csc
(
2
�
−
�
−
csc
(
�
csc
(
−
�
)
sec
(
−
�
+
sec
�{\displaystyle \sec(-\theta )=+\sec \theta }
sec
(
�
2
−
�
csc
�{\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }
sec
(
�
−
�
−
sec
�
sec
(
3
�
2
−
�
−
csc
�
sec
(
2
�
−
�
+
sec
(
�
sec
(
−
�
)
cot
(
−
�
−
cot
�{\displaystyle \cot(-\theta )=-\cot \theta }
cot
(
�
2
−
�
tan
�{\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }
cot
(
�
−
�
−
cot
�
cot
(
3
�
2
−
�
+
tan
�
cot
(
2
�
−
�
−
cot
(
�
cot
(
−
�
)
Shifts and periodicity
edit
Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
Transformation of coordinates (a,b) when shifting the angle
�{\displaystyle \theta } in increments of
�
2
{\displaystyle {\frac {\pi }{2}}}.
Shift by one quarter period Shift by one half period Shift by full periods[4] Period
sin
(
�
±
�
2
±
cos
�{\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }
sin
(
�
+
�
−
sin
�{\displaystyle \sin(\theta +\pi )=-\sin \theta }
sin
(
�
+
�
⋅
2
�
+
sin
�
2
�
cos
(
�
±
�
2
∓
sin
�{\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }
cos
(
�
+
�
−
cos
�{\displaystyle \cos(\theta +\pi )=-\cos \theta }
cos
(
�
+
�
⋅
2
�
+
cos
�
2
�
csc
(
�
±
�
2
±
sec
�{\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }
csc
(
�
+
�
−
csc
�{\displaystyle \csc(\theta +\pi )=-\csc \theta }
csc
(
�
+
�
⋅
2
�
+
csc
�
2
�
sec
(
�
±
�
2
∓
csc
�{\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }
sec
(
�
+
�
−
sec
�{\displaystyle \sec(\theta +\pi )=-\sec \theta }
sec
(
�
+
�
⋅
2
�
+
sec
�
2
�
tan
(
�
±
�
4
tan
�
±
1
1
∓
tan
�{\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}
tan
(
�
+
�
2
−
cot
�{\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }
tan
(
�
+
�
⋅
�
+
tan
�
�
cot
(
�
±
�
4
cot
�
∓
1
1
±
cot
�{\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}
cot
(
�
+
�
2
−
tan
�{\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }
cot
(
�
+
�
⋅
�
+
cot
�
�
Signs
edit
The sign of trigonometric functions depends on quadrant of the angle. If
−
�
<
�
≤
� and sgn is the sign function,
sgn
(
sin
�
sgn
(
csc
�
{
+
1
if
0
<
�
<
�
−
1
if
−
�
<
�
<
0
0
if
�
∈
{
0
,
�
}
sgn
(
cos
�
sgn
(
sec
�
{
+
1
if
−
1
2
�
<
�
<
1
2
�
−
1
if
−
�
<
�
<
−
1
2
�
or
1
2
�
<
�
<
�
0
if
�
∈
{
−
1
2
�
,
1
2
�
}
sgn
(
tan
�
sgn
(
cot
�
{
+
1
if
−
�
<
�
<
−
1
2
�
or
0
<
�
<
1
2
�
−
1
if
−
1
2
�
<
�
<
0
or
1
2
�
<
�
<
�
0
if
�
∈
{
−
1
2
�
,
0
,
1
2
�
,
�
}
The trigonometric functions are periodic with common period
2
�
,
so for values of θ outside the interval
(
−
�
,
�
]
,
they take repeating values (see § Shifts and periodicity above).
Angle sum and difference identities
edit
See also: Proofs of trigonometric identities § Angle sum identities, and Small-angle approximation § Angle sum and difference
Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.
Diagram showing the angle difference identities for
sin
(
�
−
�
)
{\displaystyle \sin(\alpha -\beta )} and
cos
(
�
−
�
)
{\displaystyle \cos(\alpha -\beta )}.
These are also known as the angle addition and subtraction theorems (or formulae).
sin
(
�
+
�
sin
�
cos
�
+
cos
�
sin
�
sin
(
�
−
�
sin
�
cos
�
−
cos
�
sin
�
cos
(
�
+
�
cos
�
cos
�
−
sin
�
sin
�
cos
(
�
−
�
cos
�
cos
�
+
sin
�
sin
�
{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}
The angle difference identities for
sin
(
�
−
�
)
{\displaystyle \sin(\alpha -\beta )} and
cos
(
�
−
�
)
{\displaystyle \cos(\alpha -\beta )} can be derived from the angle sum versions by substituting
−
�{\displaystyle -\beta } for
�{\displaystyle \beta } and using the facts that
sin
(
−
�
−
sin
(
�
)
{\displaystyle \sin(-\beta )=-\sin(\beta )} and
cos
(
−
�
cos
(
�
)
{\displaystyle \cos(-\beta )=\cos(\beta )}. They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.
These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.
Sine
sin
(
�
±
�
)
{\displaystyle =}
sin
�
cos
�
±
cos
�
sin
�{\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }[5][6]
Cosine
cos
(
�
±
�
)
{\displaystyle =}
cos
�
cos
�
∓
sin
�
sin
�{\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }[6][7]
Tangent
tan
(
�
±
�
)
{\displaystyle =}
tan
�
±
tan
�
1
∓
tan
�
tan
�{\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}[6][8]
Cosecant
csc
(
�
±
�
)
{\displaystyle =}
sec
�
sec
�
csc
�
csc
�
sec
�
csc
�
±
csc
�
sec
�{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}[9]
Secant
sec
(
�
±
�
)
{\displaystyle =}
sec
�
sec
�
csc
�
csc
�
csc
�
csc
�
∓
sec
�
sec
�{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}[9]
Cotangent
cot
(
�
±
�
)
{\displaystyle =}
cot
�
cot
�
∓
1
cot
�
±
cot
�{\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}[6][10]
Arcsine
arcsin
�
±
arcsin
�
{\displaystyle =}
arcsin
(
�
1
−
�
2
±
�
1
−
�
2
)
{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)}[11]
Arccosine
arccos
�
±
arccos
�
{\displaystyle =}
arccos
(
�
�
∓
(
1
−
�
2
)
(
1
−
�
2
)
)
{\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}[12]
Arctangent
arctan
�
±
arctan
�
{\displaystyle =}
arctan
(
�
±
�
1
∓
�
�
)
{\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}[13]
Arccotangent
arccot
�
±
arccot
�
{\displaystyle =}
arccot
(
�
�
∓
1
�
±
�
)
{\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}
Sines and cosines of sums of infinitely many angles
edit
When the series
∑
1
∞
�
�
{\textstyle \sum _{i=1}^{\infty }\theta _{i}} converges absolutely then
sin
(
∑
1
∞
�
�
∑
odd
�
≥
1
(
−
1
)
�
−
1
2
∑
�
⊆
{
1
,
2
,
3
,
…
}
|
�
�
(
∏
�
∈
�
sin
�
�
∏
�
∉
�
cos
�
�
)
cos
(
∑
1
∞
�
�
∑
even
�
≥
0
(
−
1
)
�
2
∑
�
⊆
{
1
,
2
,
3
,
…
}
|
�
�
(
∏
�
∈
�
sin
�
�
∏
�
∉
�
cos
�
�
)
.
{\displaystyle {\begin{aligned}{\sin }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggl )}&=\sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}!!\sum _{\begin{smallmatrix}A\subseteq {\,1,2,3,\dots \,}\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}\{\cos }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggr )}&=\sum _{{\text{even}}\ k\geq 0}(-1)^{\frac {k}{2}}\,\sum _{\begin{smallmatrix}A\subseteq {\,1,2,3,\dots \,}\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}.\end{aligned}}}
Because the series
∑
1
∞
�
�
{\textstyle \sum _{i=1}^{\infty }\theta _{i}} converges absolutely, it is necessarily the case that
lim
�
→
∞
�
0
,
{\textstyle \lim _{i\to \infty }\theta _{i}=0,}
lim
�
→
∞
sin
�
0
,
{\textstyle \lim _{i\to \infty }\sin \theta _{i}=0,} and
lim
�
→
∞
cos
�
1.
In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.
When only finitely many of the angles
�
�
are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
Tangents and cotangents of sums
edit
Let
�
�
(for
0
,
1
,
2
,
3
,
… ) be the kth-degree elementary symmetric polynomial in the variables
�
�tan
�
�
for
�0
,
1
,
2
,
3
,
…
,
that is,
�
1
�
∑
�
�
∑
�
tan
�
�
�
∑
�
<
�
�
�
�
∑
�
<
�
tan
�
�
tan
�
�
�
∑
�
<
�
<
�
�
�
�
�
�
∑
�
<
�
<
�
tan
�
�
tan
�
�
tan
�
�
⋮
⋮
Then
tan
(
∑
�
�
�
sin
(
∑
�
�
�
)
/
∏
�
cos
�
�
cos
(
∑
�
�
�
)
/
∏
�
cos
�
∑
odd
�
≥
1
(
−
1
)
�
−
1
2
∑
�
⊆
{
1
,
2
,
3
,
…
}
|
�
�
∏
�
∈
�
tan
�
�
∑
even
�
≥
0
(
−
1
)
�
2
∑
�
⊆
{
1
,
2
,
3
,
…
}
|
�
�
∏
�
∈
�
tan
�
�
1
−
�
3
+
�
5
−
⋯
�
0
−
�
2
+
�
4
−
⋯
cot
(
∑
�
�
�
�
0
−
�
2
+
�
4
−
⋯
�
1
−
�
3
+
�
5
−
⋯
using the sine and cosine sum formulae above.
The number of terms on the right side depends on the number of terms on the left side.
For example:
tan
(
�
1
+
�
2
�
1
�
0
−
�
�
1
+
�
2
1
−
�
1
�
tan
�
1
+
tan
�
2
1
−
tan
�
1
tan
�
2
,
tan
(
�
1
+
�
2
+
�
3
�
1
−
�
3
�
0
−
�
(
�
1
+
�
2
+
�
3
)
−
(
�
1
�
2
�
3
)
1
−
(
�
1
�
2
+
�
1
�
3
+
�
2
�
3
)
,
tan
(
�
1
+
�
2
+
�
3
+
�
4
�
1
−
�
3
�
0
−
�
2
+
�
(
�
1
+
�
2
+
�
3
+
�
4
)
−
(
�
1
�
2
�
3
+
�
1
�
2
�
4
+
�
1
�
3
�
4
+
�
2
�
3
�
4
)
1
−
(
�
1
�
2
+
�
1
�
3
+
�
1
�
4
+
�
2
�
3
+
�
2
�
4
+
�
3
�
4
)
+
(
�
1
�
2
�
3
�
4
)
,
and so on. The case of only finitely many terms can be proved by mathematical induction.[14]
Secants and cosecants of sums
edit
sec
(
∑
�
�
�
∏
�
sec
�
�
�
0
−
�
2
+
�
4
−
⋯
csc
(
∑
�
�
�
∏
�
sec
�
�
�
1
−
�
3
+
�
5
−
⋯
where
�
�
is the kth-degree elementary symmetric polynomial in the n variables
�
tan
�
�
,
1
,
…
,
�
,
and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[15] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,
sec
(
�
+
�
+
�
sec
�
sec
�
sec
�
1
−
tan
�
tan
�
−
tan
�
tan
�
−
tan
�
tan
�
csc
(
�
+
�
+
�
sec
�
sec
�
sec
�
tan
�
+
tan
�
+
tan
�
−
tan
�
tan
�
tan
�
.
Ptolemy's theorem
edit
Main article: Ptolemy's theorem
See also: History of trigonometry § Classical antiquity
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral
�
�
�
�
, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[16] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
By Thales's theorem,
∠
�
�
�
and
∠
�
�
�
are both right angles. The right-angled triangles
�
�
�
and
�
�
�
both share the hypotenuse
�
�
¯ of length 1. Thus, the side
�
�
sin
� ,
�
�
cos
� ,
�
�
sin
� and
�
�
cos
� .
By the inscribed angle theorem, the central angle subtended by the chord
�
�
¯ at the circle's center is twice the angle
∠
�
�
�
, i.e.
2
(
�
+
�
)
. Therefore, the symmetrical pair of red triangles each has the angle
�
+
� at the center. Each of these triangles has a hypotenuse of length
1
2
, so the length of
�
�
¯ is
2
×
1
2
sin
(
�
+
�
)
, i.e. simply
sin
(
�
+
�
)
. The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also
sin
(
�
+
�
)
.
When these values are substituted into the statement of Ptolemy's theorem that
|
�
�
¯
|
⋅
|
�
�
¯
�
¯
|
⋅
|
�
�
¯
|
+
|
�
�
¯
|
⋅
|
�
�
¯
|
, this yields the angle sum trigonometric identity for sine:
sin
(
�
+
�
sin
�
cos
�
+
cos
�
sin
� . The angle difference formula for
sin
(
�
−
�
)
can be similarly derived by letting the side
�
�
¯ serve as a diameter instead of
�
�
¯ .[16]
Multiple-angle and half-angle formulae
edit
Tn is the nth Chebyshev polynomial
cos
(
�
�
�
�
(
cos
�
)
{\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}[17]
de Moivre's formula, i is the imaginary unit
cos
(
�
�
)
+
�
sin
(
�
�
(
cos
�
+
�
sin
�
)
�
{\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}[18]
Multiple-angle formulae
edit
Double-angle formulae
edit
Visual demonstration of the double-angle formula for sine. The area,
1
/
2
× base × height, of an isosceles triangle is calculated, first when upright, and then on its side. When upright, the area =
sin
�
cos
�{\displaystyle \sin \theta \cos \theta }. When on its side, the area =
1
2
sin
2
�{\textstyle {\frac {1}{2}}\sin 2\theta }. Rotating the triangle does not change its area, so these two expressions are equal. Therefore,
sin
2
2
sin
�
cos
�
.
{\displaystyle \sin 2\theta =2\sin \theta \cos \theta .}
Formulae for twice an angle.[19]
sin
(
2
�
2
sin
�
cos
(
sin
�
+
cos
�
)
2
−
2
tan
�
1
+
tan
2
�{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}
cos
(
2
�
cos
2
�
−
sin
2
2
cos
2
�
−
1
−
2
sin
2
1
−
tan
2
�
1
+
tan
2
�{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}}
tan
(
2
�
2
tan
�
1
−
tan
2
�{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}
cot
(
2
�
cot
2
�
−
1
2
cot
1
−
tan
2
�
2
tan
�{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {1-\tan ^{2}\theta }{2\tan \theta }}}
sec
(
2
�
sec
2
�
2
−
sec
2
1
+
tan
2
�
1
−
tan
2
�{\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta }}}
csc
(
2
�
sec
�
csc
�
1
+
tan
2
�
2
tan
�{\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}={\frac {1+\tan ^{2}\theta }{2\tan \theta }}}
Triple-angle formulae
edit
Formulae for triple angles.[19]
sin
(
3
�
3
sin
�
−
4
sin
3
4
sin
�
sin
(
�
3
−
�
)
sin
(
�
3
+
�
)
{\displaystyle \sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)}
cos
(
3
�
4
cos
3
�
−
3
cos
4
cos
�
cos
(
�
3
−
�
)
cos
(
�
3
+
�
)
{\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)}
tan
(
3
�
3
tan
�
−
tan
3
�
1
−
3
tan
2
tan
�
tan
(
�
3
−
�
)
tan
(
�
3
+
�
)
{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)}
cot
(
3
�
3
cot
�
−
cot
3
�
1
−
3
cot
2
�{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}
sec
(
3
�
sec
3
�
4
−
3
sec
2
�{\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}}
csc
(
3
�
csc
3
�
3
csc
2
�
−
4
{\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}}
Multiple-angle formulae
edit
Formulae for multiple angles.[20]
sin
(
�
�
∑
�
odd
(
−
1
)
�
−
1
2
(
�
�
)
cos
�
−
�
�
sin
�
sin
�
∑
0
(
�
+
1
)
/
2
∑
0
�
(
−
1
)
�
−
�
(
�
2
�
+
1
)
(
�
�
)
cos
�
−
2
(
�
−
�
)
−
1
2
(
�
−
1
)
∏
0
�
−
1
sin
(
�
�
/
�
+
�
)
{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}}
cos
(
�
�
∑
�
even
(
−
1
)
�
2
(
�
�
)
cos
�
−
�
�
sin
�
∑
�0
�
/
2
∑
0
�
(
−
1
)
�
−
�
(
�
2
�
)
(
�
�
)
cos
�
−
2
(
�
−
�
)
�{\displaystyle \cos(n\theta )=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta }
cos
(
(
2
�
+
1
)
�
(
−
1
)
�
2
2
�
∏
0
2
�
cos
(
�
�
/
(
2
�
+
1
)
−
�
)
{\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}
cos
(
2
�
�
(
−
1
)
�
2
2
�
−
1
∏
0
2
�
−
1
cos
(
(
1
+
2
�
)
�
/
(
4
�
)
−
�
)
{\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}
tan
(
�
�
∑
�
odd
(
−
1
)
�
−
1
2
(
�
�
)
tan
�
�
∑
�
even
(
−
1
)
�
2
(
�
�
)
tan
�
�{\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}}
Chebyshev method
edit
The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the
(
�
−
1
)
{\displaystyle (n-1)}th and
(
�
−
2
)
{\displaystyle (n-2)}th values.[21]
cos
(
�
�
)
can be computed from
cos
(
(
�
−
1
)
�
)
,
cos
(
(
�
−
2
)
�
)
, and
cos
(
�
)
with
cos
(
�
�
2
cos
�
cos
(
(
�
−
1
)
�
)
−
cos
(
(
�
−
2
)
�
)
.
This can be proved by adding together the formulae
cos
(
(
�
−
1
)
�
+
�
cos
(
(
�
−
1
)
�
)
cos
�
−
sin
(
(
�
−
1
)
�
)
sin
�
cos
(
(
�
−
1
)
�
−
�
cos
(
(
�
−
1
)
�
)
cos
�
+
sin
(
(
�
−
1
)
�
)
sin
�
It follows by induction that
cos
(
�
�
)
is a polynomial of
cos
�
,
the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.
Similarly,
sin
(
�
�
)
can be computed from
sin
(
(
�
−
1
)
�
)
,
sin
(
(
�
−
2
)
�
)
,
and
cos
�
with
sin
(
�
�
2
cos
�
sin
(
(
�
−
1
)
�
)
−
sin
(
(
�
−
2
)
�
)
This can be proved by adding formulae for
sin
(
(
�
−
1
)
�
+
�
)
and
sin
(
(
�
−
1
)
�
−
�
)
.
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
tan
(
�
�
tan
(
(
�
−
1
)
�
)
+
tan
�
1
−
tan
(
(
�
−
1
)
�
)
tan
�
.
Half-angle formulae
edit
sin
�
sgn
(
sin
�
2
)
1
−
cos
�
2
cos
�
sgn
(
cos
�
2
)
1
+
cos
�
2
tan
�
1
−
cos
�
sin
sin
�
1
+
cos
csc
�
−
cot
tan
�
1
+
sec
sgn
(
sin
�
)
1
−
cos
�
1
+
cos
−
1
+
sgn
(
cos
�
)
1
+
tan
2
�
tan
�
cot
�
1
+
cos
�
sin
sin
�
1
−
cos
csc
�
+
cot
sgn
(
sin
�
)
1
+
cos
�
1
−
cos
�
sec
�
sgn
(
cos
�
2
)
2
1
+
cos
�
csc
�
sgn
(
sin
�
2
)
2
1
−
cos
�
[22][23]
Also
tan
�
±
�
sin
�
±
sin
�
cos
�
+
cos
�
tan
(
�
2
+
�
4
sec
�
+
tan
�
1
−
sin
�
1
+
sin
−
tan
�
2
|
|
1
+
tan
�
2
|
Table
edit
See also: Tangent half-angle formula
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
Sine Cosine Tangent Cotangent
Double-angle formula[24][25]
sin
(
2
�
2
sin
�
cos
�
=
2
tan
�
1
+
tan
2
�
cos
(
2
�
cos
2
�
−
sin
2
2
cos
2
�
−
1
−
2
sin
2
1
−
tan
2
�
1
+
tan
2
�
tan
(
2
�
2
tan
�
1
−
tan
2
�
cot
(
2
�
cot
2
�
−
1
2
cot
�
Triple-angle formula[17][26]
sin
(
3
�
−
sin
3
�
+
3
cos
2
�
sin
−
4
sin
3
�
+
3
sin
�
cos
(
3
�
cos
3
�
−
3
sin
2
�
cos
4
cos
3
�
−
3
cos
�
tan
(
3
�
3
tan
�
−
tan
3
�
1
−
3
tan
2
�
cot
(
3
�
3
cot
�
−
cot
3
�
1
−
3
cot
2
�
Half-angle formula[22][23]
sin
�
sgn
(
sin
�
2
)
1
−
cos
�
2
(
or
sin
2
�
1
−
cos
�
2
)
cos
�
sgn
(
cos
�
2
)
1
+
cos
�
2
(
or
cos
2
�
1
+
cos
�
2
)
tan
�
csc
�
−
cot
±
1
−
cos
�
1
+
cos
sin
�
1
+
cos
1
−
cos
�
sin
�
tan
�
+
�
sin
�
+
sin
�
cos
�
+
cos
�
tan
(
�
2
+
�
4
sec
�
+
tan
�
1
−
sin
�
1
+
sin
−
tan
�
2
|
|
1
+
tan
�
2
|
tan
�
tan
�
1
+
1
+
tan
2
�
for
�
∈
(
−
�
2
,
�
2
)
cot
�
csc
�
+
cot
±
1
+
cos
�
1
−
cos
sin
�
1
−
cos
1
+
cos
�
sin
�
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.[citation needed]
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where
�
is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.
Power-reduction formulae
edit
Obtained by solving the second and third versions of the cosine double-angle formula.
Sine Cosine Other
sin
2
1
−
cos
(
2
�
)
2
{\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}
cos
2
1
+
cos
(
2
�
)
2
sin
2
�
cos
2
1
−
cos
(
4
�
)
8
sin
3
3
sin
�
−
sin
(
3
�
)
4
{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}}
cos
3
3
cos
�
+
cos
(
3
�
)
4
sin
3
�
cos
3
3
sin
(
2
�
)
−
sin
(
6
�
)
32
sin
4
3
−
4
cos
(
2
�
)
+
cos
(
4
�
)
8
{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}}
cos
4
3
+
4
cos
(
2
�
)
+
cos
(
4
�
)
8
sin
4
�
cos
4
3
−
4
cos
(
4
�
)
+
cos
(
8
�
)
128
sin
5
10
sin
�
−
5
sin
(
3
�
)
+
sin
(
5
�
)
16
{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}}
cos
5
10
cos
�
+
5
cos
(
3
�
)
+
cos
(
5
�
)
16
sin
5
�
cos
5
10
sin
(
2
�
)
−
5
sin
(
6
�
)
+
sin
(
10
�
)
512
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse
�
�
¯{\displaystyle {\overline {AD}}} of the blue triangle has length
2
cos
�{\displaystyle 2\cos \theta }. The angle
∠
�
�
�
{\displaystyle \angle DAE} is
�{\displaystyle \theta }, so the base
�
�
¯{\displaystyle {\overline {AE}}} of that triangle has length
2
cos
2
�{\displaystyle 2\cos ^{2}\theta }. That length is also equal to the summed lengths of
�
�
¯{\displaystyle {\overline {BD}}} and
�
�
¯{\displaystyle {\overline {AF}}}, i.e.
1
+
cos
(
2
�
)
{\displaystyle 1+\cos(2\theta )}. Therefore,
2
cos
2
1
+
cos
(
2
�
)
{\displaystyle 2\cos ^{2}\theta =1+\cos(2\theta )}. Dividing both sides by
2
{\displaystyle 2} yields the power-reduction formula for cosine:
cos
2
{\displaystyle \cos ^{2}\theta =}
1
2
(
1
+
cos
(
2
�
)
)
{\textstyle {\frac {1}{2}}(1+\cos(2\theta ))}. The half-angle formula for cosine can be obtained by replacing
�{\displaystyle \theta } with
�
/
2
{\displaystyle \theta /2} and taking the square-root of both sides:
cos
(
�
/
2
±
(
1
+
cos
�
)
/
2
.
{\textstyle \cos \left(\theta /2\right)=\pm {\sqrt {\left(1+\cos \theta \right)/2}}.}
Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle
�
�
�
{\displaystyle EBD} are all right-angled and similar, and all contain the angle
�{\displaystyle \theta }. The hypotenuse
�
�
¯{\displaystyle {\overline {BD}}} of the red-outlined triangle has length
2
sin
�{\displaystyle 2\sin \theta }, so its side
�
�
¯{\displaystyle {\overline {DE}}} has length
2
sin
2
�{\displaystyle 2\sin ^{2}\theta }. The line segment
�
�
¯{\displaystyle {\overline {AE}}} has length
cos
2
�{\displaystyle \cos 2\theta } and sum of the lengths of
�
�
¯{\displaystyle {\overline {AE}}} and
�
�
¯{\displaystyle {\overline {DE}}} equals the length of
�
�
¯{\displaystyle {\overline {AD}}}, which is 1. Therefore,
cos
2
�
+
2
sin
2
1
{\displaystyle \cos 2\theta +2\sin ^{2}\theta =1}. Subtracting
cos
2
�{\displaystyle \cos 2\theta } from both sides and dividing by 2 by two yields the power-reduction formula for sine:
sin
2
{\displaystyle \sin ^{2}\theta =}
1
2
(
1
−
cos
(
2
�
)
)
. The half-angle formula for sine can be obtained by replacing
� with
�
/
2
and taking the square-root of both sides:
sin
(
�
/
2
±
(
1
−
cos
�
)
/
2
.
Note that this figure also illustrates, in the vertical line segment
�
�
¯ , that
sin
2
2
sin
�
cos
� .
In general terms of powers of
sin
� or
cos
� the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem.
if n is ...
cos
�
�
sin
�
�
n is odd
cos
�
2
2
�
∑
0
�
−
1
2
(
�
�
)
cos
(
(
�
−
2
�
)
�
)
sin
�
2
2
�
∑
0
�
−
1
2
(
−
1
)
(
�
−
1
2
−
�
)
(
�
�
)
sin
(
(
�
−
2
�
)
�
)
n is even
cos
�
1
2
�
(
�
�
2
)
+
2
2
�
∑
0
�
2
−
1
(
�
�
)
cos
(
(
�
−
2
�
)
�
)
sin
�
1
2
�
(
�
�
2
)
+
2
2
�
∑
0
�
2
−
1
(
−
1
)
(
�
2
−
�
)
(
�
�
)
cos
(
(
�
−
2
�
)
�
)
Product-to-sum and sum-to-product identities
edit
The product-to-sum identities[27] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations.[28] See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.
Product-to-sum identities
edit
cos
�
cos
cos
(
�
−
�
)
+
cos
(
�
+
�
)
2
{\displaystyle \cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}}
sin
�
sin
cos
(
�
−
�
)
−
cos
(
�
+
�
)
2
{\displaystyle \sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}}
sin
�
cos
sin
(
�
+
�
)
+
sin
(
�
−
�
)
2
{\displaystyle \sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}}
cos
�
sin
sin
(
�
+
�
)
−
sin
(
�
−
�
)
2
{\displaystyle \cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}}
tan
�
tan
cos
(
�
−
�
)
−
cos
(
�
+
�
)
cos
(
�
−
�
)
+
cos
(
�
+
�
)
{\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}
tan
�
cot
sin
(
�
+
�
)
+
sin
(
�
−
�
)
sin
(
�
+
�
)
−
sin
(
�
−
�
)
{\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}}
∏
1
�
cos
�
1
2
�
∑
�
∈
�
cos
(
�
1
�
1
+
⋯
+
�
�
�
�
)
where
(
�
1
,
…
,
�
�
)
∈
{
1
,
−
1
}
�
{\displaystyle {\begin{aligned}\prod {k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e{1}\theta {1}+\cdots +e{n}\theta {n})\[6pt]&{\text{where }}e=(e{1},\ldots ,e_{n})\in S={1,-1}^{n}\end{aligned}}}
∏
1
�
sin
�
(
−
1
)
⌊
�
2
⌋
2
�
{
∑
�
∈
�
cos
(
�
1
�
1
+
⋯
+
�
�
�
�
)
∏
1
�
�
�
if
�
is even
,
∑
�
∈
�
sin
(
�
1
�
1
+
⋯
+
�
�
�
�
)
∏
1
�
�
�
if
�
is odd
{\displaystyle \prod {k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e{1}\theta {1}+\cdots +e{n}\theta {n})\prod _{j=1}^{n}e{j}\;{\text{if}}\;n\;{\text{is even}},\\displaystyle \sum {e\in S}\sin(e{1}\theta {1}+\cdots +e{n}\theta {n})\prod _{j=1}^{n}e{j}\;{\text{if}}\;n\;{\text{is odd}}\end{cases}}}
Sum-to-product identities
edit
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle
� and the red right-angled triangle has angle
� . Both have a hypotenuse of length 1. Auxiliary angles, here called
�
and
�
, are constructed such that
(
�
+
�
)
/
2
and
(
�
−
�
)
/
2
. Therefore,
�
+
�
and
�
−
�
. This allows the two congruent purple-outline triangles
�
�
�
and
�
�
�
to be constructed, each with hypotenuse
cos
�
and angle
�
at their base. The sum of the heights of the red and blue triangles is
sin
�
+
sin
� , and this is equal to twice the height of one purple triangle, i.e.
2
sin
�
cos
�
. Writing
�
and
�
in that equation in terms of
� and
� yields a sum-to-product identity for sine:
sin
�
+
sin
2
sin
(
�
+
�
2
)
cos
(
�
−
�
2
)
. Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.
The sum-to-product identities are as follows:[29]
sin
�
±
sin
2
sin
(
�
±
�
2
)
cos
(
�
∓
�
2
)
cos
�
+
cos
2
cos
(
�
+
�
2
)
cos
(
�
−
�
2
)
cos
�
−
cos
−
2
sin
(
�
+
�
2
)
sin
(
�
−
�
2
)
tan
�
±
tan
sin
(
�
±
�
)
cos
�
cos
�
Hermite's cotangent identity
edit
Main article: Hermite's cotangent identity
Charles Hermite demonstrated the following identity.[30] Suppose
�
1
,
…
,
�
�
are complex numbers, no two of which differ by an integer multiple of π. Let
�
�
,
∏
1
≤
�
≤
�
�
≠
�
cot
(
�
�
−
�
�
)
(in particular,
�
1
,
1
,
being an empty product, is 1). Then
cot
(
�
−
�
1
)
⋯
cot
(
�
−
�
�
cos
�
�
2
+
∑
1
�
�
�
,
�
cot
(
�
−
�
�
)
.
The simplest non-trivial example is the case n = 2:
cot
(
�
−
�
1
)
cot
(
�
−
�
2
−
1
+
cot
(
�
1
−
�
2
)
cot
(
�
−
�
1
)
+
cot
(
�
2
−
�
1
)
cot
(
�
−
�
2
)
.
Finite products of trigonometric functions
edit
For coprime integers n, m
∏
�1
�
(
2
�
+
2
cos
(
2
�
�
�
�
+
�
)
2
(
�
�
(
�
)
+
(
−
1
)
�
+
�
cos
(
�
�
)
)
where Tn is the Chebyshev polynomial.[citation needed]
The following relationship holds for the sine function
∏
�1
�
−
1
sin
(
�
�
�
�
2
�
−
1
.
More generally for an integer n > 0[31]
sin
(
�
�
2
�
−
1
∏
0
�
−
1
sin
(
�
�
�
+
�
2
�
−
1
∏
1
�
sin
(
�
�
�
−
�
)
.
or written in terms of the chord function
crd
�
≡
2
sin
1
2
�
,
crd
(
�
�
∏
�1
�
crd
(
�
�
2
�
−
�
)
.
This comes from the factorization of the polynomial
�
�
−
1
into linear factors (cf. root of unity): For a point z on the complex unit circle and an integer n > 0,
�
�
−
∏
�1
�
�
−
exp
(
�
�
2
�
�
)
.
Linear combinations
edit
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of
�
{\displaystyle c} and
�{\displaystyle \varphi }.
Sine and cosine
edit
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[32][33]
�
cos
�
+
�
sin
�
cos
(
�
+
�
)
{\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}
where
�
{\displaystyle c} and
�{\displaystyle \varphi } are defined as so:
sgn
(
�
)
�
2
+
�
2
,
arctan
(
−
�
/
�
)
,
{\displaystyle {\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}},\\varphi &={\arctan }{\bigl (}{-b/a}{\bigr )},\end{aligned}}}
given that
�
≠
0.
{\displaystyle a\neq 0.}
Arbitrary phase shift
edit
More generally, for arbitrary phase shifts, we have
�
sin
(
�
+
�
�
)
+
�
sin
(
�
+
�
�
�
sin
(
�
+
�
)
{\displaystyle a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )}
where
�
{\displaystyle c} and
�{\displaystyle \varphi } satisfy:
�
2�
2
+
�
2
+
2
�
�
cos
(
�
�
−
�
�
)
,
tan
�
sin
�
�
+
�
sin
�
�
�
cos
�
�
+
�
cos
�
�
.
More than two sinusoids
edit
See also: Phasor addition
The general case reads[33]
∑
�
�
�
sin
(
�
+
�
�
�
sin
(
�
+
�
)
,
where
�
∑
�
,
�
�
�
�
�
cos
(
�
�
−
�
�
)
and
tan
∑
�
�
�
sin
�
�
∑
�
�
�
cos
�
�
.
Lagrange's trigonometric identities
edit
These identities, named after Joseph Louis Lagrange, are:[34][35][36]
∑
�0
�
sin
�
cos
1
2
�
−
cos
(
(
�
+
1
2
)
�
)
2
sin
1
2
�
∑
0
�
cos
�
sin
1
2
�
+
sin
(
(
�
+
1
2
)
�
)
2
sin
1
2
�
{\displaystyle {\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2}}\theta -\cos \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2}}\theta +\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\end{aligned}}}
for
�
≢
0
(
mod
2
�
)
.
{\displaystyle \theta \not \equiv 0{\pmod {2\pi }}.}
A related function is the Dirichlet kernel:
�
�
(
�
1
+
2
∑
1
�
cos
�
sin
(
(
�
+
1
2
)
�
)
sin
1
2
�
.
{\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{\sin {\tfrac {1}{2}}\theta }}.}
A similar identity is[37]
∑
�1
�
cos
(
2
�
−
1
)
sin
(
2
�
�
)
2
sin
�
.
{\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha }}.}
The proof is the following. By using the angle sum and difference identities,
sin
(
�
+
�
)
−
sin
(
�
−
�
2
cos
�
sin
�
.
{\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.}
Then let's examine the following formula,
2
sin
�
∑
1
�
cos
(
2
�
−
1
)
2
sin
�
cos
�
+
2
sin
�
cos
3
�
+
2
sin
�
cos
5
�
+
…
+
2
sin
�
cos
(
2
�
−
1
)
�
{\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha }
and this formula can be written by using the above identity,
2
sin
�
∑
1
�
cos
(
2
�
−
1
)
∑
�1
�
(
sin
(
2
�
�
)
−
sin
(
2
(
�
−
1
)
�
)
(
sin
2
�
−
sin
0
)
+
(
sin
4
�
−
sin
2
�
)
+
(
sin
6
�
−
sin
4
�
)
+
…
+
(
sin
(
2
�
�
)
−
sin
(
2
(
�
−
1
)
�
)
sin
(
2
�
�
)
.
So, dividing this formula with
2
sin
� completes the proof.
Certain linear fractional transformations
edit
If
�
(
�
)
{\displaystyle f(x)} is given by the linear fractional transformation
�
(
�
(
cos
�
)
�
−
sin
�
(
sin
�
)
�
+
cos
�
,
{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}
and similarly
�
(
�
(
cos
�
)
�
−
sin
�
(
sin
�
)
�
+
cos
�
,
{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}
then
�
(
�
(
�
)
�
(
�
(
�
)
(
cos
(
�
+
�
)
)
�
−
sin
(
�
+
�
)
(
sin
(
�
+
�
)
)
�
+
cos
(
�
+
�
)
.
{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}
More tersely stated, if for all
�{\displaystyle \alpha } we let
�
�{\displaystyle f_{\alpha }} be what we called
�
{\displaystyle f} above, then
�
�
∘
�
�
�
+
�
.
{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.}
If
�
{\displaystyle x} is the slope of a line, then
�
(
�
)
{\displaystyle f(x)} is the slope of its rotation through an angle of
−
�
.
{\displaystyle -\alpha .}
Relation to the complex exponential function
edit
Main article: Euler's formula
Euler's formula states that, for any real number x:[38]
�
�
cos
�
+
�
sin
�
,
{\displaystyle e^{ix}=\cos x+i\sin x,}
where i is the imaginary unit. Substituting −x for x gives us:
�
−
�
cos
(
−
�
)
+
�
sin
(
−
�
cos
�
−
�
sin
�
.
{\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}
These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,[39][40]
cos
�
�
�
+
�
−
�
�
2
{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}
sin
�
�
�
−
�
−
�
�
2
�
{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}
These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = eiθ eiφ means that
cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.
Function Inverse function[41]
sin
�
�
�
−
�
−
�
�
2
�
arcsin
−
�
ln
(
�
�
+
1
−
�
2
)
cos
�
�
�
+
�
−
�
�
2
arccos
−
�
ln
(
�
+
�
2
−
1
)
tan
−
�
�
�
�
−
�
−
�
�
�
�
�
+
�
−
�
�
arctan
�
2
ln
(
�
+
�
�
−
�
)
csc
2
�
�
�
�
−
�
−
�
�
arccsc
−
�
ln
(
�
�
+
1
−
1
�
2
)
sec
2
�
�
�
+
�
−
�
�
arcsec
−
�
ln
(
1
�
+
�
1
−
1
�
2
)
cot
�
�
�
�
+
�
−
�
�
�
�
�
−
�
−
�
�
arccot
�
2
ln
(
�
−
�
�
+
�
)
cis
�
�
�
arccis
−
�
ln
�
Series expansion
edit
When using a power series expansion to define trigonometric functions, the following identities are obtained:[42]
sin
�
−
�
3
3
!
+
�
5
5
!
−
�
7
7
!
∑
�0
∞
(
−
1
)
�
�
2
�
+
1
(
2
�
+
1
)
!
,
{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},}
cos
1
−
�
2
2
!
+
�
4
4
!
−
�
6
6
!
+
∑
�0
∞
(
−
1
)
�
�
2
�
(
2
�
)
!
.
{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.}
Infinite product formulae
edit
For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[43][44]
sin
�
∏
1
∞
(
1
−
�
2
�
2
�
2
)
,
cos
∏
�1
∞
(
1
−
�
2
�
2
(
�
−
1
2
)
2
)
,
sinh
�
∏
1
∞
(
1
+
�
2
�
2
�
2
)
,
cosh
∏
�1
∞
(
1
+
�
2
�
2
(
�
−
1
2
)
2
)
.
{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)!{\vphantom {)}}^{2}}}\right),\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)!{\vphantom {)}}^{2}}}\right).\end{aligned}}}
Inverse trigonometric functions
edit
Main article: Inverse trigonometric functions
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[45]
sin
(
arcsin
�
�
cos
(
arcsin
�
1
−
�
2
tan
(
arcsin
�
�
1
−
�
2
sin
(
arccos
�
1
−
�
2
cos
(
arccos
�
�
tan
(
arccos
�
1
−
�
2
�
sin
(
arctan
�
�
1
+
�
2
cos
(
arctan
�
1
1
+
�
2
tan
(
arctan
�
�
sin
(
arccsc
�
1
�
cos
(
arccsc
�
�
2
−
1
�
tan
(
arccsc
�
1
�
2
−
1
sin
(
arcsec
�
�
2
−
1
�
cos
(
arcsec
�
1
�
tan
(
arcsec
�
�
2
−
1
sin
(
arccot
�
1
1
+
�
2
cos
(
arccot
�
�
1
+
�
2
tan
(
arccot
�
1
�
{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\end{aligned}}}
Taking the multiplicative inverse of both sides of the each equation above results in the equations for
1
sin
,
1
cos
,
and
1
tan
.
{\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ and }}\cot ={\frac {1}{\tan }}.} The right hand side of the formula above will always be flipped. For example, the equation for
cot
(
arcsin
�
)
{\displaystyle \cot(\arcsin x)} is:
cot
(
arcsin
�
1
tan
(
arcsin
�
1
�
1
−
�
1
−
�
2
�
{\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}}
while the equations for
csc
(
arccos
�
)
{\displaystyle \csc(\arccos x)} and
sec
(
arccos
�
)
{\displaystyle \sec(\arccos x)} are:
csc
(
arccos
�
1
sin
(
arccos
�
1
1
−
�
2
and
sec
(
arccos
�
1
cos
(
arccos
�
1
�
.
{\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ and }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.}
The following identities are implied by the reflection identities. They hold whenever
�
,
�
,
�
,
−
�
,
−
�
,
and
−
�
are in the domains of the relevant functions.
�
2
=
arcsin
(
�
)
+
arccos
(
�
)
=
arctan
(
�
)
+
arccot
(
�
)
=
arcsec
(
�
)
+
arccsc
(
�
)
�
=
arccos
(
�
)
+
arccos
(
−
�
)
=
arccot
(
�
)
+
arccot
(
−
�
)
=
arcsec
(
�
)
+
arcsec
(
−
�
)
0
=
arcsin
(
�
)
+
arcsin
(
−
�
)
=
arctan
(
�
)
+
arctan
(
−
�
)
=
arccsc
(
�
)
+
arccsc
(
−
�
)
Also,[46]
arctan
�
+
arctan
1
{
�
2
,
if
�
0
−
�
2
,
if
�
<
0
arccot
�
+
arccot
1
{
�
2
,
if
�
0
3
�
2
,
if
�
<
0
arccos
1
arcsec
�
and
arcsec
1
arccos
�
arcsin
1
arccsc
�
and
arccsc
1
arcsin
�
The arctangent function can be expanded as a series:[47]
arctan
(
�
�
∑
�1
�
arctan
�
1
+
(
�
−
1
)
�
�
2
Identities without variables
edit
In terms of the arctangent function we have[46]
arctan
1
arctan
1
3
+
arctan
1
7
.
{\displaystyle \arctan {\frac {1}{2}}=\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}
The curious identity known as Morrie's law,
cos
20
∘
⋅
cos
40
∘
⋅
cos
80
1
8
,
{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},}
is a special case of an identity that contains one variable:
∏
�0
�
−
1
cos
(
2
�
�
sin
(
2
�
�
)
2
�
sin
�
.
{\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.}
Similarly,
sin
20
∘
⋅
sin
40
∘
⋅
sin
80
3
8
{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}
is a special case of an identity with
20
∘{\displaystyle x=20^{\circ }}:
sin
�
⋅
sin
(
60
∘
−
�
)
⋅
sin
(
60
∘
+
�
sin
3
�
4
.
{\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.}
For the case
15
∘{\displaystyle x=15^{\circ }},
sin
15
∘
⋅
sin
45
∘
⋅
sin
75
2
8
,
sin
15
∘
⋅
sin
75
1
4
.
{\displaystyle {\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2}}{8}},\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}
For the case
10
∘ ,
sin
10
∘
⋅
sin
50
∘
⋅
sin
70
1
8
.
The same cosine identity is
cos
�
⋅
cos
(
60
∘
−
�
)
⋅
cos
(
60
∘
+
�
cos
3
�
4
.
Similarly,
cos
10
∘
⋅
cos
50
∘
⋅
cos
70
3
8
,
cos
15
∘
⋅
cos
45
∘
⋅
cos
75
2
8
,
cos
15
∘
⋅
cos
75
1
4
.
Similarly,
tan
50
∘
⋅
tan
60
∘
⋅
tan
70
tan
80
∘
,
tan
40
∘
⋅
tan
30
∘
⋅
tan
20
tan
10
∘
.
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
cos
24
∘
+
cos
48
∘
+
cos
96
∘
+
cos
168
1
2
.
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
cos
2
�
21
+
cos
(
2
⋅
2
�
21
)
+
cos
(
4
⋅
2
�
21
)
+
cos
(
5
⋅
2
�
21
)
+
cos
(
8
⋅
2
�
21
)
+
cos
(
10
⋅
2
�
21
1
2
.
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than
21
/
2
that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Other cosine identities include:[48]
2
cos
�
1
,
2
cos
�
5
×
2
cos
2
�
1
,
2
cos
�
7
×
2
cos
2
�
7
×
2
cos
3
�
1
,
and so forth for all odd numbers, and hence
cos
�
3
+
cos
�
5
×
cos
2
�
5
+
cos
�
7
×
cos
2
�
7
×
cos
3
�
7
+
1.
Many of those curious identities stem from more general facts like the following:[49]
∏
�1
�
−
1
sin
�
�
�
2
�
−
1
and
∏
1
�
−
1
cos
�
�
sin
�
�
2
2
�
−
1
.
Combining these gives us
∏
�1
�
−
1
tan
�
�
�
sin
�
�
2
If n is an odd number (
�2
�
+
1
) we can make use of the symmetries to get
∏
�1
�
tan
�
�
2
�
+
2
�
+
1
The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
∏
�1
�
sin
(
2
�
−
1
)
�
4
∏
�1
�
cos
(
2
�
−
1
)
�
4
2
2
�
Computing π
edit
An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula:
�
44
arctan
1
5
−
arctan
1
239
or, alternatively, by using an identity of Leonhard Euler:
�
5
arctan
1
7
+
2
arctan
3
79
or by using Pythagorean triples:
�arccos
4
5
+
arccos
5
13
+
arccos
16
arcsin
3
5
+
arcsin
12
13
+
arcsin
63
65
.
Others include:[50][46]
�
4arctan
1
2
+
arctan
1
3
,
arctan
1
+
arctan
2
+
arctan
3
,
�
42
arctan
1
3
+
arctan
1
7
.
Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1
k=1 arctan tk ∈ (π/4, 3π/4), let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,
�
2∑
�1
�
arctan
(
�
�
)
∑
�1
�
sgn
(
�
�
)
arccos
(
1
−
�
�
2
1
+
�
�
2
)
∑
�1
�
arcsin
(
2
�
�
1
+
�
�
2
)
∑
�1
�
arctan
(
2
�
�
1
−
�
�
2
)
,
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the tk values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2).
For example, for n = 3 terms,
�
2arctan
(
�
�
)
+
arctan
(
�
�
)
+
arctan
(
�
�
−
�
�
�
�
+
�
�
)
for any a, b, c, d > 0.
An identity of Euclid
edit
Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
sin
2
18
∘
+
sin
2
30
sin
2
36
∘
.
Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.
Composition of trigonometric functions
edit
These identities involve a trigonometric function of a trigonometric function:[51]
cos
(
�
sin
�
�
0
(
�
)
+
2
∑
1
∞
�
2
�
(
�
)
cos
(
2
�
�
)
{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum {k=1}^{\infty }J{2k}(t)\cos(2kx)}
sin
(
�
sin
�
2
∑
0
∞
�
2
�
+
1
(
�
)
sin
(
(
2
�
+
1
)
�
)
{\displaystyle \sin(t\sin x)=2\sum {k=0}^{\infty }J{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}
cos
(
�
cos
�
�
0
(
�
)
+
2
∑
1
∞
(
−
1
)
�
�
2
�
(
�
)
cos
(
2
�
�
)
{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum {k=1}^{\infty }(-1)^{k}J{2k}(t)\cos(2kx)}
sin
(
�
cos
�
2
∑
0
∞
(
−
1
)
�
�
2
�
+
1
(
�
)
cos
(
(
2
�
+
1
)
�
)
{\displaystyle \sin(t\cos x)=2\sum {k=0}^{\infty }(-1)^{k}J{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}
where Ji are Bessel functions.
Further "conditional" identities for the case α + β + γ = 180°
edit
The following formulae apply to arbitrary plane triangles and follow from
�
+
�
+
180
∘
,
{\displaystyle \alpha +\beta +\gamma =180^{\circ },} as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).
tan
�
+
tan
�
+
tan
tan
�
tan
�
tan
�
cot
�
cot
�
+
cot
�
cot
�
+
cot
�
cot
�
cot
(
�
2
)
+
cot
(
�
2
)
+
cot
(
�
2
cot
(
�
2
)
cot
(
�
2
)
cot
(
�
2
)
tan
(
�
2
)
tan
(
�
2
)
+
tan
(
�
2
)
tan
(
�
2
)
+
tan
(
�
2
)
tan
(
�
2
)
sin
�
+
sin
�
+
sin
4
cos
(
�
2
)
cos
(
�
2
)
cos
(
�
2
)
−
sin
�
+
sin
�
+
sin
4
cos
(
�
2
)
sin
(
�
2
)
sin
(
�
2
)
cos
�
+
cos
�
+
cos
4
sin
(
�
2
)
sin
(
�
2
)
sin
(
�
2
)
+
1
−
cos
�
+
cos
�
+
cos
4
sin
(
�
2
)
cos
(
�
2
)
cos
(
�
2
)
−
1
sin
(
2
�
)
+
sin
(
2
�
)
+
sin
(
2
�
4
sin
�
sin
�
sin
�
−
sin
(
2
�
)
+
sin
(
2
�
)
+
sin
(
2
�
4
sin
�
cos
�
cos
�
cos
(
2
�
)
+
cos
(
2
�
)
+
cos
(
2
�
−
4
cos
�
cos
�
cos
�
−
1
−
cos
(
2
�
)
+
cos
(
2
�
)
+
cos
(
2
�
−
4
cos
�
sin
�
sin
�
+
1
sin
2
�
+
sin
2
�
+
sin
2
2
cos
�
cos
�
cos
�
+
2
−
sin
2
�
+
sin
2
�
+
sin
2
2
cos
�
sin
�
sin
�
cos
2
�
+
cos
2
�
+
cos
2
−
2
cos
�
cos
�
cos
�
+
1
−
cos
2
�
+
cos
2
�
+
cos
2
−
2
cos
�
sin
�
sin
�
+
1
sin
2
(
2
�
)
+
sin
2
(
2
�
)
+
sin
2
(
2
�
−
2
cos
(
2
�
)
cos
(
2
�
)
cos
(
2
�
)
+
2
cos
2
(
2
�
)
+
cos
2
(
2
�
)
+
cos
2
(
2
�
2
cos
(
2
�
)
cos
(
2
�
)
cos
(
2
�
)
+
1
sin
2
(
�
2
)
+
sin
2
(
�
2
)
+
sin
2
(
�
2
)
+
2
sin
(
�
2
)
sin
(
�
2
)
sin
(
�
2
)
{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \1&=\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta \\cot \left({\frac {\alpha }{2}}\right)+\cot \left({\frac {\beta }{2}}\right)+\cot \left({\frac {\gamma }{2}}\right)&=\cot \left({\frac {\alpha }{2}}\right)\cot \left({\frac {\beta }{2}}\right)\cot \left({\frac {\gamma }{2}}\right)\1&=\tan \left({\frac {\beta }{2}}\right)\tan \left({\frac {\gamma }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\alpha }{2}}\right)\tan \left({\frac {\beta }{2}}\right)\\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)\\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)+1\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)-1\\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \sin \beta \sin \gamma \-\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \cos \beta \cos \gamma \\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \cos \beta \cos \gamma -1\-\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \sin \beta \sin \gamma +1\\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )&=-2\cos(2\alpha )\cos(2\beta )\cos(2\gamma )+2\\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )&=2\cos(2\alpha )\,\cos(2\beta )\,\cos(2\gamma )+1\1&=\sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)\end{aligned}}}
Historical shorthands
edit
Main articles: Versine and Exsecant
The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
Miscellaneous
edit
Dirichlet kernel
edit
Main article: Dirichlet kernel
The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:
1
+
2
cos
�
+
2
cos
(
2
�
)
+
2
cos
(
3
�
)
+
⋯
+
2
cos
(
�
�
sin
(
(
�
+
1
2
)
�
)
sin
(
1
2
�
)
.
{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.}
The convolution of any integrable function of period
2
�{\displaystyle 2\pi } with the Dirichlet kernel coincides with the function's
�
{\displaystyle n}th-degree Fourier approximation. The same holds for any measure or generalized function.
Tangent half-angle substitution
edit
Main article: Tangent half-angle substitution
If we set
tan
�
2
,
{\displaystyle t=\tan {\frac {x}{2}},}
then[52]
sin
2
�
1
+
�
2
;
cos
1
−
�
2
1
+
�
2
;
�
�
1
+
�
�
1
−
�
�
{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}}}
where
�
�
cos
�
+
�
sin
�
,
{\displaystyle e^{ix}=\cos x+i\sin x,} sometimes abbreviated to cis x.
When this substitution of
�
for tan
x
/
2
is used in calculus, it follows that
sin
�
is replaced by
2t
/
1 + t2
,
cos
�
is replaced by
1 − t2
/
1 + t2
and the differential dx is replaced by
2 dt
/
1 + t2
. Thereby one converts rational functions of
sin
�
and
cos
�
to rational functions of
�
in order to find their antiderivatives.
Viète's infinite product
edit
See also: Viète's formula and Sinc function
cos
�
2
⋅
cos
�
4
⋅
cos
�
8
∏
�1
∞
cos
�
2
sin
�
sinc
�
.
See also
edit
Aristarchus's inequality
Derivatives of trigonometric functions
Exact trigonometric values (values of sine and cosine expressed in surds)
Exsecant
Half-side formula
Hyperbolic function
Laws for solution of triangles:
Law of cosines
Spherical law of cosines
Law of sines
Law of tangents
Law of cotangents
Mollweide's formula
List of integrals of trigonometric functions
Mnemonics in trigonometry
Pentagramma mirificum
Proofs of trigonometric identities
Prosthaphaeresis
Pythagorean theorem
Tangent half-angle formula
Trigonometric number
Trigonometry
Uses of trigonometry
Versine and haversine
References
edit
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 4, eqn 4.3.45". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 73. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Selby 1970, p. 188
Abramowitz and Stegun, p. 72, 4.3.13–15
Abramowitz and Stegun, p. 72, 4.3.7–9
Abramowitz and Stegun, p. 72, 4.3.16
Weisstein, Eric W. "Trigonometric Addition Formulas". MathWorld.
Abramowitz and Stegun, p. 72, 4.3.17
Abramowitz and Stegun, p. 72, 4.3.18
"Angle Sum and Difference Identities". http://www.milefoot.com. Retrieved 2019-10-12.
Abramowitz and Stegun, p. 72, 4.3.19
Abramowitz and Stegun, p. 80, 4.4.32
Abramowitz and Stegun, p. 80, 4.4.33
Abramowitz and Stegun, p. 80, 4.4.34
Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York: ACM. pp. 207–211. doi:10.1145/74540.74566. ISBN 0-89791-325-6.
Hardy, Michael (2016). "On Tangents and Secants of Infinite Sums". American Mathematical Monthly. 123 (7): 701–703. doi:10.4169/amer.math.monthly.123.7.701.
"Sine, Cosine, and Ptolemy's Theorem".
Weisstein, Eric W. "Multiple-Angle Formulas". MathWorld.
Abramowitz and Stegun, p. 74, 4.3.48
Selby 1970, pg. 190
Weisstein, Eric W. "Multiple-Angle Formulas". mathworld.wolfram.com. Retrieved 2022-02-06.
Ward, Ken. "Multiple angles recursive formula". Ken Ward's Mathematics Pages.
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 4, eqn 4.3.20-22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 72. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Weisstein, Eric W. "Half-Angle Formulas". MathWorld.
Abramowitz and Stegun, p. 72, 4.3.24–26
Weisstein, Eric W. "Double-Angle Formulas". MathWorld.
Abramowitz and Stegun, p. 72, 4.3.27–28
Abramowitz and Stegun, p. 72, 4.3.31–33
Eves, Howard (1990). An introduction to the history of mathematics (6th ed.). Philadelphia: Saunders College Pub. p. 309. ISBN 0-03-029558-0. OCLC 20842510.
Abramowitz and Stegun, p. 72, 4.3.34–39
Johnson, Warren P. (Apr 2010). "Trigonometric Identities à la Hermite". American Mathematical Monthly. 117 (4): 311–327. doi:10.4169/000298910x480784. S2CID 29690311.
"Product Identity Multiple Angle".
Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
Weisstein, Eric W. "Harmonic Addition Theorem". MathWorld.
Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics. 21 (2): 140. Bibcode:1953AmJPh..21..140M. doi:10.1119/1.1933371.
Agarwal, Ravi P.; O'Regan, Donal (2008). Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems (illustrated ed.). Springer Science & Business Media. p. 185. ISBN 978-0-387-79146-3. Extract of page 185
Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". Handbook of Mathematical Formulas and Integrals (4th ed.). Academic Press. ISBN 978-0-12-374288-9.
Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151.
Abramowitz and Stegun, p. 74, 4.3.47
Abramowitz and Stegun, p. 71, 4.3.2
Abramowitz and Stegun, p. 71, 4.3.1
Abramowitz and Stegun, p. 80, 4.4.26–31
Abramowitz and Stegun, p. 74, 4.3.65–66
Abramowitz and Stegun, p. 75, 4.3.89–90
Abramowitz and Stegun, p. 85, 4.5.68–69
Abramowitz & Stegun 1972, p. 73, 4.3.45
Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
S. M. Abrarov, R. K. Jagpal, R. Siddiqui and B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", Mathematics, 9 (17), 2162, arXiv:2107.01027, doi:10.3390/math9172162
Humble, Steve (Nov 2004). "Grandma's identity". Mathematical Gazette. 88: 524–525. doi:10.1017/s0025557200176223. S2CID 125105552.
Weisstein, Eric W. "Sine". MathWorld.
Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39.
Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972, formulae 9.1.42–9.1.45
Abramowitz and Stegun, p. 72, 4.3.23
Bibliography
edit
Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0.
Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York: Barnes & Noble, LCCN 61-9103
Selby, Samuel M., ed. (1970), Standard Mathematical Tables (18th ed.), The Chemical Rubber Co.
External links
edit
Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5+
5
/
8
°, and for the same angles csc and sec and tan
Complete List of Trigonometric Formulas
Last edited 15 days ago by DVdm
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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN YASEEN 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Timothy Jude Smith (born 9 June 1982[1]), known by his stage name Timmy Trumpet, is an Australian musician, DJ, songwriter and record producer. He has become known internationally for playing the trumpet live and making use of jazz elements in dance music.
Timmy Trumpet
Timmy Trumpet during an Interview at Airbeat One in 2019
Timmy Trumpet during an Interview at Airbeat One in 2019
Background information
Birth name
Timothy Jude Smith
Born
9 June 1982 (age 41)
Sydney, Australia
Genres
Electro housebig room houseMelbourne bouncepsytrancehardstylehardcore
Occupation(s)
Musician, DJ, songwriter, record producer
Instrument(s)
TrumpetDigital TrumpetPiano
Years active
2005–present
Labels
Spinnin'RevealedMinistry of Sound,MonstercatDim MakArmadaUltra MusicSINPHONYDharmaDoorn
Website
Official website
Timmy Trumpet's breakthrough single "Freaks" with New Zealand rapper Savage was certified gold by the RIAA, six times platinum by ARIA, and triple platinum by Recorded Music NZ. Timmy Trumpet is currently ranked the number 6 DJ in the world, according to the DJ Mag Top 100.[2]
On 25 March 2019, Timmy Trumpet became the first trumpet player to perform in zero gravity, a project partnership between the European Space Agency and BigCityBeats.[3][4]
Early life
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Timothy Jude Smith was born in Sydney, Australia. He started playing the trumpet at 4 years old, taught by his father. Smith studied at Carlingford High School for his secondary education, where he was the school captain. At thirteen, Smith was named 'Young Musician of the Year' before being granted a full scholarship to the Conservatorium of Music, where he was tutored by Anthony Heinrich of the Sydney Symphony Orchestra. Within two years he secured a position as the leading solo trumpet player in the Australian All-Star Stage Band.[5]
Career
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2000s–2015: Early career in Australia
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Smith adopted the stage name Timmy Trumpet and began performing as a trumpeter alongside Australian DJ duo the Stafford Brothers at Australian nightclubs and music festivals. He taught himself how to DJ and began playing solo shows billed as Timmy Trumpet, incorporating live trumpet into his DJ sets.[6][7][8]
Timmy Trumpet signed early music releases to Australian record labels OneLove, Central Station and Hussle Recordings, and produced various mix compilations for Ministry of Sound Australia which included national promotional tours.[9][10][11]
In 2011, Timmy Trumpet was selected as the support act for Stevie Wonder at The Star Casino in Sydney's grand opening.[12] Timmy also appeared as himself in a reoccurring role on The Stafford Brothers, a reality TV show originally broadcast on FOX8.
Timmy Trumpet completed one of his first international tours in 2014 alongside Australian DJ's Will Sparks and Joel Fletcher for their Bounce Bus Tour in North America.[13]
Between 2012 and 2015, Timmy had a number of singles hit number one on the ARIA Club Chart, the first being a collaboration titled 'Sassafras', with Melbourne DJ Chardy. Sassafras was the number two track on the ARIA end of year Charts - Top 50 Club Tracks of 2012.[14] 'The Buzz', a collaboration with New World Sound reached number one on Beatport in addition to reaching number one on the ARIA Club Chart. The Buzz was also used in the Dirty Grandpa official TV trailer.[15] "Freaks", a collaboration with Savage reached number one of ARIA Australian Singles, Dance and Club Charts and has since been accredited six times platinum in Australia. Freaks is the highest selling single of all time on Ministry of Sound Australia.[16]
Timmy Trumpet was voted the number one DJ in Australia at the 2015 inthemix Awards.[17]
2015–2016: "Freaks" and international attention
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"Freaks" is the twelfth single release for Timmy Trumpet. The track garnered international success charting in Australia, New Zealand, Hungary, Poland, Belgium, France and the Netherlands. The single sold more than one million copies worldwide and has been streamed online over five hundred million times. "Freaks" reached number three on the ARIA chart and number one in New Zealand.
"Freaks" features in the Hollywood TV shows and films Santa Clarita Diet, Collide, Teen Wolf, American Ultra and Training Day.[18]
"Freaks" debuted on the Billboard emerging artists + Twitter chart at number nine when a Vine titled ‘When mama isn't home’ went viral. The video features Brisbane father and son Russ and Toby Bauer. The father plays Freaks on a trombone whilst his son slams the oven door repeatedly.[19] As of October 2015, the Vine received over 26 million loops, 201,800 likes and 158,200 Re-Vines. On 18 November 2015, Russ and Toby Bauer remade the viral video live on Australian national TV on Nine Network's Today where Timmy Trumpet surprised them, turning up halfway through the performance.[20]
By 2015, Timmy Trumpet was gaining recognition within the music industry overseas and started writing and producing music with international artists. "Toca", the first of a series of international collaborations, was with Guatemalan-American DJ and producer, Carnage and Indian-American DJ and producer KSHMR. "Toca" is a Big room rework of the classical Johann Sebastian Bach composition "Toccata and Fugue in D minor". "Toca" was released on 26 June 2015 on Ultra Music. The track has since become a trademark of Timmy Trumpet's live performances, utilizing elements of the original orchestral introduction and playing live trumpet.
In 2016, Timmy Trumpet released three singles on Dutch dance label Spinnin' Records. The first release "Psy or Die" was circulated around the DJ scene without anyone knowing who had authored the unreleased track at the time, and it was played at festivals and nightclubs by Armin Van Buuren, W&W, Blasterjaxx, NERVO and Andrew Rayel.[21] It was later revealed when released that "Psy or Die" was by Timmy Trumpet and Carnage and the track was popular on the European dance music festival scene that summer. On 27 September 2016 Spinnin Records released "Party Till We Die" by MAKJ and Timmy Trumpet featuring rock artist and motivational speaker Andrew W.K.[22] Timmy's debut solo single released on Spinnin' titled "Oracle" reached number one on Beatport, and has received more than 150 million online streams as of October 2019.[23][24]
2017–2018: Global music festivals
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In 2017, Timmy Trumpet made his main stage debut at Tomorrowland in Boom, Belgium. His performance was the most streamed DJ set of all time on Tomorrowland's official YouTube channel, and as of October 2019 has received twenty-two million views.[25]
Between 2017 and 2018, Timmy Trumpet became a fixture at dance music festivals all over the world, playing main stage sets at Parookaville, Electric Love, Creamfields, Airbeat One, and many more.
Timmy Trumpet collaborated with Hardwell on the track "The Underground" which was released on Revealed Recordings. He also collaborated with DJ and producers Vini Vici (formerly known as Sesto Sento) on "100" and Blasterjaxx on "Narco".
On 21 June 2018, Timmy Trumpet played a sold-out show at Ormeau Park in Belfast, Northern Ireland, selling 15,000 tickets.[26]
2019: World tour and zero gravity
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On 25 March 2019, Timmy Trumpet joined a host of celebrities on the BigCityBeats World Club Dome "Zero Gravity 2.0". A specially converted A310 aircraft, usually used by the European Space Agency for training astronauts and performing scientific experiments in microgravity and other levels of reduced gravity, took off from Frankfurt Airport. The aircraft engaged in a series of parabolic flights that caused the passengers to experience a total of six minutes of true weightlessness.
On 18 April 2019, Timmy Trumpet announced a 60-date world tour.[27] The single "World At Our Feet" was released on 10 May, along with an animated lyric video.[28] The tour included main stage headlining performances at 31 festivals across Europe, North America, South America, and Asia.[29] Timmy hosted his own stages at Tomorrowland and Creamfields. Timmy made his main stage debut at Electric Daisy Carnival in Las Vegas on 19 May,[30] and later was announced as a headliner of Electric Daisy Carnival in Orlando, South Korea, and China.
Timmy began a vlog series on YouTube chronicling his life on tour. The first episode follows his experiences at Electric Daisy Carnival in Las Vegas.[31]
In February 2021, Timmy Trumpet launched a weekly radio show and YouTube stream called "SINPHONY Radio" where songs from all forms of EDM are played. During the 42nd episode of the show, Timmy stated that the show was heard in more than 40 countries.[32]
On 21 August 2021, Timmy Trumpet collaborated with Dutch DJ Armin van Buuren on a song called "Anita."[33] The song was featured in a commercial for Cinnamon Toast Crunch a few months after its release.[34]
In 2021 and 2022, "Narco" gained popularity in Major League Baseball when New York Mets closer Edwin Díaz adopted it as his walk-on music; on 30 August 2022, Timmy attended a Mets game and met Diaz—telling him that he was "officially a Mets fan for life". During a game the following day, Timmy performed the song's intro live at Citi Field when Diaz entered the game.[35]
Musical style
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Timmy Trumpet is renowned for his energetic live performances and use of pyrotechnics on stage. Timmy plays mixed platform electronic music, trumpet, blending many sub-genres together, playing music at various speeds, and incorporating live trumpet solos over custom edits. Timmy's DJ sets predominantly feature his own original tracks, collaborations and remixes.[36]
Timmy has released music in a number of subgenres of electronic music including house, electro house, Melbourne bounce, hardstyle, psychedelic trance, big room house, and funk.
Timmy has cited Daft Punk as early inspiration to start experimenting with electronic dance music. In an interview with Belfast Telegraph, Timmy said: "I was a trumpeter before I was a DJ, I would play alongside songs I heard on the radio. I remember playing to Daft Punk for the first time. That's when I realised there might be something to this trumpet/EDM thing."[37]
Personal life
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In 2020, Timmy proposed to his Hungarian girlfriend Anett during a livestream.[38] They married in Budapest in 2022.[39]
Discography
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Main article: Timmy Trumpet discography
Mad World (2020)
References
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"Timmy Trumpet - Biography & History". AllMusic. Retrieved 27 March 2020.
"Poll 2021: Timmy Trumpet". DJMag.com. Retrieved 14 October 2021.
Graves, Ciarra (4 April 2019). "BigCityBeats' WORLD CLUB DOME Historically Touches Down with Exclusive Zero Gravity Party". The Nocturnal Times. Retrieved 17 October 2019.
""It was like no other experience I've had before": Timmy Trumpet plays DJ set in zero gravity!". Music Network. 4 April 2019. Retrieved 17 October 2019.
Millar, Mark (14 August 2019). "INTERVIEW: Timmy Trumpet – "No one parties like the Irish! The crowds in Belfast are always crazy"". XS Noize. Retrieved 17 October 2019.
"The Met presents the Stafford Brothers and Timmy Trumpet". moshtix.com.au. Retrieved 17 October 2019.
""Future Music Festival 2011" Stafford Brothers ft Timmy Trumpet". Retrieved 17 October 2019 – via YouTube.
""Future Music Festival 2012" Stafford Brothers ft Timmy Trumpet". youtube.com. Retrieved 17 October 2019.
"Timmy Trumpet Discography". discogs.com. Retrieved 17 October 2019.
Fry, Douglas (2 August 2012). "Ministry of Sound: Sessions 9". The Sydney Morning Herald. Retrieved 17 October 2019.
"Ministry of Sound: Sessions 10". Sydney Unleashed. Retrieved 17 October 2019.
"Stars Converge On The Star For Stevie Wonder". Pedestrian. 26 October 2011. Retrieved 10 January 2012.
"Will Sparks, Joel Fletcher and Timmy Trumpet join forces for US tour". itm.junkee.com. Archived from the original on 7 August 2021. Retrieved 7 August 2021.
"ARIA Charts - End Of Year Charts - Top 50 Club Tracks 2012". aria.com.au. Retrieved 17 October 2019.
"'The Buzz' by Timmy Trumpet & New World Sound in Dirty Grandpa Trailer". spinninrecords.com. Retrieved 17 October 2019.
"Universal Music Publishing Group Biography". umusicpub.com. Retrieved 17 October 2019.
Cunningham, Katie (25 June 2015). "The 2015 inthemix Awards Results Are In". Ibis Play. Retrieved 17 October 2019.
"Timmy Trumpet music featured in movies and TV shows". tunefind.com. Retrieved 18 October 2019.
Reid, Poppy (27 October 2015). "Case Study: Timmy Trumpet's dual US chart debut". Music Network. Retrieved 17 October 2019.
"Timmy Trumpet joins Oven Boy for a TODAY Show jam". youtube.com. Retrieved 17 October 2019.
Bein, Kat (7 July 2016). "Carnage & Timmy Trumpet Release 'Psy or Die,' and You Might Have Heard it Already". billboard.com. Retrieved 19 October 2019.
Spanos, Brittany (27 September 2016). "Hear Andrew W.K.'s Aggressive EDM Debut 'Party Til We Die'". rollingstone.com. Retrieved 19 October 2019.
"Timmy Trumpet - Oracle (Official Music Video)". youtube.com. Retrieved 17 October 2019.
"Timmy Trumpet - Oracle on Spotify". Spotify. Retrieved 17 October 2019.
"Tomorrowland Belgium 2017". Retrieved 17 October 2019 – via YouTube.
"Timmy Trumpet sells out Ormeau Park". belfasttelegraph.co.uk. 22 June 2018. Retrieved 19 October 2019.
"Timmy Trumpet World Tour Announcement". instagram.com. Archived from the original on 24 December 2021. Retrieved 18 October 2019.
"Timmy Trumpet - World At Our Feet (Official Lyric Video)". youtube.com. Retrieved 18 October 2019.
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"EDC Las Vegas Announces Full Lineup for 2019 Edition". edm.com. 27 March 2019. Retrieved 18 October 2019.
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SINPHONY Radio w/ Timmy Trumpet | Episode 042, retrieved 19 December 2021
Jasmin (30 August 2021). "Armin van Buuren and Timmy Trumpet Collaborate for 'Anita'". EDMTunes. Retrieved 19 December 2021.
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"Timmy Trumpet Gets Engaged To His Girlfriend On Livestream [VIDEO]". T.H.E - Music Essentials. 12 July 2020. Retrieved 12 February 2022.
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Coulomb's law
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Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law[1] of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the electrostatic force or Coulomb force.[2] Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the theory of electromagnetism and maybe even its starting point,[1] as it allowed meaningful discussions of the amount of electric charge in a particle.[3]
The magnitude of the electrostatic force F between two point charges q1 and q2 is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges attract each other.
The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them.[4] Coulomb discovered that bodies with like electrical charges repel:
It follows therefore from these three tests, that the repulsive force that the two balls – [that were] electrified with the same kind of electricity – exert on each other, follows the inverse proportion of the square of the distance.[5]
Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:
=
�
e
|
�
1
|
|
�
2
|
�
2
{\displaystyle |F|=k_{\text{e}}{\frac {|q_{1}||q_{2}|}{r^{2}}}}
Here, ke is a constant, q1 and q2 are the quantities of each charge, and the scalar r is the distance between the charges.
The force is along the straight line joining the two charges. If the charges have the same sign, the electrostatic force between them makes them repel; if they have different signs, the force between them makes them attract.
Being an inverse-square law, the law is similar to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces always make things attract, while electrostatic forces make charges attract or repel. Also, gravitational forces are much weaker than electrostatic forces.[2] Coulomb's law can be used to derive Gauss's law, and vice versa. In the case of a single point charge at rest, the two laws are equivalent, expressing the same physical law in different ways.[6] The law has been tested extensively, and observations have upheld the law on the scale from 10−16 m to 108 m.[6]
History
edit
Charles-Augustin de Coulomb
Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper. Thales of Miletus made the first recorded description of static electricity around 600 BC,[7] when he noticed that friction could make a piece of amber attract small objects.[8][9]
In 1600, English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber.[8] He coined the Neo-Latin word electricus ("of amber" or "like amber", from ἤλεκτρον [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.[10] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.[11]
Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did (i.e., as the inverse square of the distance) included Daniel Bernoulli[12] and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Franz Aepinus who supposed the inverse-square law in 1758.[13]
Based on experiments with electrically charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this.[14] In 1767, he conjectured that the force between charges varied as the inverse square of the distance.[15][16]
Coulomb's torsion balance
In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as x−2.06.[17]
In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England.[18] In his notes, Cavendish wrote, "We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the 2 +
1
/
50
th and that of the 2 −
1
/
50
th, and there is no reason to think that it differs at all from the inverse duplicate ratio".
Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism.[4] He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak torsion spring. In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread. The ball was charged with a known charge of static electricity, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law.
Scalar form
edit
Coulomb's law can be stated as a simple mathematical expression. The scalar form gives the magnitude of the vector of the electrostatic force F between two point charges q1 and q2, but not its direction. If r is the distance between the charges, the magnitude of the force is
=
|
�
1
�
2
|
4
�
�
0
�
2
,
{\displaystyle |\mathbf {F} |={\frac {|q_{1}q_{2}|}{4\pi \varepsilon _{0}r^{2}}},}
where ε0 is the electric constant. If the product q1q2 is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.[19]
Vector form
edit
In the image, the vector F1 is the force experienced by q1, and the vector F2 is the force experienced by q2. When q1q2 > 0 the forces are repulsive (as in the image) and when q1q2 < 0 the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.
Coulomb's law in vector form states that the electrostatic force
�
1
{\textstyle \mathbf {F} {1}} experienced by a charge,
�
1
{\displaystyle q{1}} at position
�
1
{\displaystyle \mathbf {r} {1}}, in the vicinity of another charge,
�
2
{\displaystyle q{2}} at position
�
2
{\displaystyle \mathbf {r} _{2}}, in a vacuum is equal to[20]
�
1�
1
�
2
4
�
�
0
�
1
−
�
2
|
�
1
−
�
2
|
�
1
�
2
4
�
�
0
�
^
12
|
�
12
|
2
{\displaystyle \mathbf {F} {1}={\frac {q{1}q_{2}}{4\pi \varepsilon {0}}}{\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|^{3}}}={\frac {q{1}q_{2}}{4\pi \varepsilon _{0}}}{\frac {\mathbf {\hat {r}} _{12}}{|\mathbf {r} _{12}|^{2}}}}
where
�
�
1
−
�
2
{\textstyle {\boldsymbol {r}}{12}={\boldsymbol {r}}{1}-{\boldsymbol {r}}_{2}} is the vectorial distance between the charges,
�
^
�
12
|
�
12
|
{\textstyle {\widehat {\mathbf {r} }}{12}={\frac {\mathbf {r} _{12}}{|\mathbf {r} _{12}|}}} a unit vector pointing from
�
2
{\textstyle q{2}} to
�
1
{\textstyle q_{1}}, and
�
0
{\displaystyle \varepsilon _{0}} the electric constant. Here,
�
^
12
{\textstyle \mathbf {\hat {r}} _{12}} is used for the vector notation.
The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the unit vector,
�
^
12
{\textstyle {\widehat {\mathbf {r} }}{12}}, parallel with the line from charge
�
2
{\displaystyle q{2}} to charge
�
1
{\displaystyle q_{1}}.[21] If both charges have the same sign (like charges) then the product
�
1
�
2
is positive and the direction of the force on
�
1
is given by
�
^
12
; the charges repel each other. If the charges have opposite signs then the product
�
1
�
2
is negative and the direction of the force on
�
1
is
−
�
^
12
; the charges attract each other.
The electrostatic force
�
2
experienced by
�
2
, according to Newton's third law, is
�
−
�
1
.
System of discrete charges
edit
The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed.
Force
�
on a small charge
�
at position
�
, due to a system of
�
discrete charges in vacuum is[20]
�
(
�
�
4
�
�
0
∑
1
�
�
�
�
−
�
�
|
�
−
�
�
|
�
4
�
�
0
∑
1
�
�
�
�
^
�
|
�
�
|
2
,
where
�
�
and
�
�
are the magnitude and position respectively of the ith charge,
�
^
�
is a unit vector in the direction of
�
�
−
�
�
, a vector pointing from charges
�
�
to
�
.[21]
Continuous charge distribution
edit
In this case, the principle of linear superposition is also used. For a continuous charge distribution, an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge
�
�
. The distribution of charge is usually linear, surface or volumetric.
For a linear charge distribution (a good approximation for charge in a wire) where
�
(
�
′
)
gives the charge per unit length at position
�
′
, and
�
ℓ
′
is an infinitesimal element of length,[22]
�
�
�
(
�
′
)
�
ℓ
′
.
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where
�
(
�
′
)
gives the charge per unit area at position
�
′
, and
�
�
′
is an infinitesimal element of area,
�
�
�
(
�
′
)
�
�
′
.
For a volume charge distribution (such as charge within a bulk metal) where
�
(
�
′
)
gives the charge per unit volume at position
�
′
, and
�
�
′
is an infinitesimal element of volume,[21]
�
�
�
(
�
′
)
�
�
′
.
The force on a small test charge
�
at position
�
in vacuum is given by the integral over the distribution of charge
�
(
�
�
4
�
�
0
∫
�
�
′
�
−
�
′
|
�
−
�
′
|
3
.
The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which
|
�
−
�
′
0
because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. Charge is always discrete in reality, and the "continuous charge" assumption is just an approximation that is not supposed to allow
|
�
−
�
′
0
to be analyzed.
Coulomb constant
edit
The Coulomb constant is a proportionality factor that appears in Coulomb's law and related formulas. Denoted
�
e
{\displaystyle k_{\text{e}}}, it is also called the electric force constant or electrostatic constant[23] hence the subscript 'e'. The Coulomb constant is given by
�
1
4
�
�
0
{\textstyle k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}}. The constant
�
0
{\displaystyle \varepsilon _{0}} is the vacuum electric permittivity (also known as the electric constant).[24] It should not be confused with
�
�
{\displaystyle \varepsilon _{r}}, which is the dimensionless relative permittivity of the material in which the charges are immersed, or with their product
�
�
0
�
�
{\displaystyle \varepsilon _{a}=\varepsilon _{0}\varepsilon _{r}}, which is called "absolute permittivity of the material" and is still used in electrical engineering.
Since the 2019 redefinition of the SI base units,[25][26] the Coulomb constant, as calculated from CODATA 2018 recommended values, is[27]
�
e8.987
551
792
3
(
14
)
×
10
9
N
⋅
m
2
⋅
C
−
2
.
{\displaystyle k_{\text{e}}=8.987\,551\,792\,3\,(14)\times 10^{9}\ \mathrm {N{\cdot }m^{2}{\cdot }C^{-2}} .}
Limitations
edit
There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:[28]
The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
The charges must not overlap (e.g. they must be distinct point charges).
The charges must be stationary with respect to a nonaccelerating frame of reference.
The last of these is known as the electrostatic approximation. When movement takes place, Einstein's theory of relativity must be taken into consideration, and a result, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the magnetic force, and is described by magnetic fields. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct, but when the charges are moving more quickly in relation to each other, the full electrodynamics rules (incorporating the magnetic force) must be considered.
Electric field
edit
Main article: Electric field
If two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.
An electric field is a vector field that associates to each point in space the Coulomb force experienced by a unit test charge.[20] The strength and direction of the Coulomb force
�
{\textstyle \mathbf {F} } on a charge
�
�
{\textstyle q_{t}} depends on the electric field
�
{\textstyle \mathbf {E} } established by other charges that it finds itself in, such that
�
�
�
{\textstyle \mathbf {F} =q_{t}\mathbf {E} }. In the simplest case, the field is considered to be generated solely by a single source point charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by the principle of superposition.
If the field is generated by a positive source point charge
�
, the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge
�
�
would move if placed in the field. For a negative point source charge, the direction is radially inwards.
The magnitude of the electric field E can be derived from Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field E created by a single source point charge Q at a certain distance from it r in vacuum is given by
=
�
e
|
�
|
�
2
A system N of charges
�
�
stationed at
�
�
produces an electric field whose magnitude and direction is, by superposition
�
(
�
1
4
�
�
0
∑
1
�
�
�
�
−
�
�
|
�
−
�
�
|
3
Atomic forces
edit
See also: Coulomb explosion
Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids. Generally, as the distance between ions increases, the force of attraction, and binding energy, approach zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.
Relation to Gauss's law
edit
Learn more
This article duplicates the scope of other articles, specifically Gauss's_law#Relation_to_Coulomb's_law.
Deriving Gauss's law from Coulomb's law
edit
Gauss's law can be derived from Coulomb's law and the assumption that electric field obeys the superposition principle, which says that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed in a region of space).
Outline of proof
Coulomb's law states that the electric field due to a stationary point charge is:
�
(
�
�
4
�
�
0
�
�
�
2
,
{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{r}}{r^{2}}},}
where
er is the radial unit vector,
r is the radius, |r|,
ε0 is the electric constant,
q is the charge of the particle, which is assumed to be located at the origin.
Using the expression from Coulomb's law, we get the total field at r by using an integral to sum the field at r due to the infinitesimal charge at each other point s in space, to give
�
(
�
1
4
�
�
0
∫
�
(
�
)
(
�
−
�
)
|
�
−
�
|
3
�
3
�
where ρ is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem[29]
∇
⋅
�
|
�
|
−
∇
2
1
|
�
4
�
�
(
�
)
where δ(r) is the Dirac delta function, the result is
∇
⋅
�
(
�
1
�
0
∫
�
(
�
)
�
(
�
−
�
)
�
3
�
Using the "sifting property" of the Dirac delta function, we arrive at
∇
⋅
�
(
�
�
(
�
)
�
0
,
which is the differential form of Gauss' law, as desired.
Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.
Deriving Coulomb's law from Gauss's law
edit
Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).
Outline of proof
Taking S in the integral form of Gauss' law to be a spherical surface of radius r, centered at the point charge Q, we have
∮
�
�
⋅
�
�
�
0
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
4
�
�
2
�
^
⋅
�
(
�
�
�
0
where r̂ is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get
�
(
�
�
4
�
�
0
�
^
�
2
which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss' law.
In relativity
edit
Coulomb's law can be used to gain insight into the form of the magnetic field generated by moving charges since by special relativity, in certain cases the magnetic field can be shown to be a transformation of forces caused by the electric field. When no acceleration is involved in a particle's history, Coulomb's law can be assumed on any test particle in its own inertial frame, supported by symmetry arguments in solving Maxwell's equation, shown above. Coulomb's law can be expanded to moving test particles to be of the same form. This assumption is supported by Lorentz force law which, unlike Coulomb's law is not limited to stationary test charges. Considering the charge to be invariant of observer, the electric and magnetic fields of a uniformly moving point charge can hence be derived by the Lorentz transformation of the four force on the test charge in the charge's frame of reference given by Coulomb's law and attributing magnetic and electric fields by their definitions given by the form of Lorentz force.[30] The fields hence found for uniformly moving point charges are given by:[31]
�
4
�
�
0
�
3
1
−
�
2
(
1
−
�
2
sin
2
�
)
3
/
2
�
{\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}\mathbf {r} }
�
4
�
�
0
�
3
1
−
�
2
(
1
−
�
2
sin
2
�
)
3
/
2
�
×
�
�
�
×
�
�
2
where
�
is the charge of the point source,
�
is the position vector from the point source to the point in space,
�
is the velocity vector of the charged particle,
� is the ratio of speed of the charged particle divided by the speed of light and
� is the angle between
�
and
�
.
This form of solutions need not obey Newton's third law as is the case in the framework of special relativity (yet without violating relativistic-energy momentum conservation).[32] Note that the expression for electric field reduces to Coulomb's law for non-relativistic speeds of the point charge and that the magnetic field in non-relativistic limit (approximating
�
≪
1
) can be applied to electric currents to get the Biot–Savart law. These solutions, when expressed in retarded time also correspond to the general solution of Maxwell's equations given by solutions of Liénard–Wiechert potential, due to the validity of Coulomb's law within its specific range of application. Also note that the spherical symmetry for gauss law on stationary charges is not valid for moving charges owing to the breaking of symmetry by the specification of direction of velocity in the problem. Agreement with Maxwell's equations can also be manually verified for the above two equations.[33]
Coulomb potential
edit
See also: Electric potential
Quantum field theory
edit
Learn more
This article may be too technical for most readers to understand. (October 2020)
The most basic Feynman diagram for QED interaction between two fermions
The Coulomb potential admits continuum states (with E > 0), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom.[34] It can also be derived within the non-relativistic limit between two charged particles, as follows:
Under Born approximation, in non-relativistic quantum mechanics, the scattering amplitude
�
(
|
�
⟩
→
|
�
′
⟩
)
{\textstyle {\mathcal {A}}(|\mathbf {p} \rangle \to |\mathbf {p} '\rangle )} is:
�
(
|
�
⟩
→
|
�
′
⟩
)
−
2
�
�
(
�
�
−
�
�
′
)
(
−
�
)
∫
�
3
�
�
(
�
)
�
−
�
(
�
−
�
′
)
�
This is to be compared to the:
∫
�
3
�
(
2
�
)
3
�
�
�
�
0
⟨
�
′
,
�
|
�
|
�
,
�
⟩
where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.
Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with
�
0
≫
|
�
|
⟨
�
′
,
�
|
�
|
�
,
�
⟩
|
�
�
�
−
�
�
2
|
�
−
�
′
|
2
−
�
�
(
2
�
)
2
�
(
�
�
,
�
−
�
�
′
,
�
)
(
2
�
)
4
�
(
�
−
�
′
)
Comparing with the QM scattering, we have to discard the
(
2
�
)
2
as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:
∫
�
(
�
)
�
−
�
(
�
−
�
′
)
�
�
3
�
2
|
�
−
�
′
|
2
−
�
�
where Fourier transforming both sides, solving the integral and taking
�
→
0
at the end will yield
�
(
�
�
2
4
�
�
as the Coulomb potential.[35]
However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.[36][37]
The Coulomb potential, and its derivation, can be seen as a special case of the Yukawa potential, which is the case where the exchanged boson – the photon – has no rest mass.[34]
Simple experiment to verify Coulomb's law
edit
Learn more
This section may contain an excessive amount of intricate detail that may interest only a particular audience. (October 2020)
Experiment to verify Coulomb's law.
It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass
�
{\displaystyle m} and same-sign charge
�
{\displaystyle q}, hanging from two ropes of negligible mass of length
�
{\displaystyle l}. The forces acting on each sphere are three: the weight
�
�
{\displaystyle mg}, the rope tension
�
{\displaystyle \mathbf {T} } and the electric force
�
{\displaystyle \mathbf {F} }. In the equilibrium state:
�
sin
�
�
1
{\displaystyle \mathbf {T} \sin \theta _{1}=\mathbf {F} _{1}}
(1)
and
�
cos
�
�
�
{\displaystyle \mathbf {T} \cos \theta _{1}=mg}
(2)
Dividing (1) by (2):
sin
�
1
cos
�
�
1
�
�
⇒
�
�
�
tan
�
1
{\displaystyle {\frac {\sin \theta {1}}{\cos \theta _{1}}}={\frac {F{1}}{mg}}\Rightarrow F_{1}=mg\tan \theta _{1}}
(3)
Let
�
1
{\displaystyle \mathbf {L} _{1}} be the distance between the charged spheres; the repulsion force between them
�
1
{\displaystyle \mathbf {F} _{1}}, assuming Coulomb's law is correct, is equal to
�
1�
2
4
�
�
0
�
1
2
(Coulomb's law)
so:
�
2
4
�
�
0
�
1
�
�
tan
�
1
(4)
If we now discharge one of the spheres, and we put it in contact with the charged sphere, each one of them acquires a charge
�
2
. In the equilibrium state, the distance between the charges will be
�
2
<
�
1
and the repulsion force between them will be:
�
2(
�
2
)
2
4
�
�
0
�
2
�
2
4
4
�
�
0
�
2
2
(5)
We know that
�
�
�
tan
�
2
and:
�
2
4
4
�
�
0
�
2
�
�
tan
�
2
Dividing (4) by (5), we get:
(
�
2
4
�
�
0
�
1
2
)
(
�
2
4
4
�
�
0
�
2
2
�
�
tan
�
1
�
�
tan
�
2
⇒
4
(
�
2
�
1
)
tan
�
1
tan
�
2
(6)
Measuring the angles
�
1
and
�
2
and the distance between the charges
�
1
and
�
2
is sufficient to verify that the equality is true taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently great, the angles will be small enough to make the following approximation:
tan
�
≈
sin
�
2
�
2
ℓ
⇒
tan
�
1
tan
�
2
≈
�
1
2
ℓ
�
2
2
ℓ
(7)
Using this approximation, the relationship (6) becomes the much simpler expression:
�
1
2
ℓ
�
2
2
ℓ
≈
4
(
�
2
�
1
)
2
⇒
�
1
�
2
≈
4
(
�
2
�
1
)
2
⇒
�
1
�
2
≈
4
3
(8)
In this way, the verification is limited to measuring the distance between the charges and checking that the division approximates the theoretical value.
See also
edit
icon Electronics portal
Biot–Savart law
Darwin Lagrangian
Electromagnetic force
Gauss's law
Method of image charges
Molecular modelling
Newton's law of universal gravitation, which uses a similar structure, but for mass instead of charge
Static forces and virtual-particle exchange
Casimir effect
References
edit
Huray, Paul G. (2010). Maxwell's equations. Hoboken, New Jersey: Wiley. pp. 8, 57. ISBN 978-0-470-54991-9. OCLC 739118459.
Halliday, David; Resnick, Robert; Walker, Jearl (2013). Fundamentals of Physics. John Wiley & Sons. pp. 609, 611. ISBN 9781118230718.
Roller, Duane; Roller, D. H. D. (1954). The development of the concept of electric charge: Electricity from the Greeks to Coulomb. Cambridge, Massachusetts: Harvard University Press. p. 79.
Coulomb (1785). "Premier mémoire sur l'électricité et le magnétisme" [First dissertation on electricity and magnetism]. Histoire de l'Académie Royale des Sciences in French. pp. 569–577.
Coulomb (1785). "Second mémoire sur l'électricité et le magnétisme" [Second dissertation on electricity and magnetism]. Histoire de l'Académie Royale des Sciences in French. pp. 578–611. Il résulte donc de ces trois essais, que l'action répulsive que les deux balles électrifées de la même nature d'électricité exercent l'une sur l'autre, suit la raison inverse du carré des distances.
Purcell, Edward M. (21 January 2013). Electricity and magnetism (3rd ed.). Cambridge. ISBN 9781107014022.
Cork, C.R. (2015). "Conductive fibres for electronic textiles". Electronic Textiles: 3–20. doi:10.1016/B978-0-08-100201-8.00002-3. ISBN 9780081002018.
Stewart, Joseph (2001). Intermediate Electromagnetic Theory. World Scientific. p. 50. ISBN 978-981-02-4471-2.
Simpson, Brian (2003). Electrical Stimulation and the Relief of Pain. Elsevier Health Sciences. pp. 6–7. ISBN 978-0-444-51258-1.
Baigrie, Brian (2007). Electricity and Magnetism: A Historical Perspective. Greenwood Press. pp. 7–8. ISBN 978-0-313-33358-3.
Chalmers, Gordon (1937). "The Lodestone and the Understanding of Matter in Seventeenth Century England". Philosophy of Science. 4 (1): 75–95. doi:10.1086/286445. S2CID 121067746.
Socin, Abel (1760). Acta Helvetica Physico-Mathematico-Anatomico-Botanico-Medica (in Latin). Vol. 4. Basileae. pp. 224–25.
Heilbron, J.L. (1979). Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics. Los Angeles, California: University of California Press. pp. 460–462 and 464 (including footnote 44). ISBN 978-0486406886.
Schofield, Robert E. (1997). The Enlightenment of Joseph Priestley: A Study of his Life and Work from 1733 to 1773. University Park: Pennsylvania State University Press. pp. 144–56. ISBN 978-0-271-01662-7.
Priestley, Joseph (1767). The History and Present State of Electricity, with Original Experiments. London, England. p. 732.
Elliott, Robert S. (1999). Electromagnetics: History, Theory, and Applications. Wiley. ISBN 978-0-7803-5384-8. Archived from the original on 2014-03-10. Retrieved 2009-10-17.
Robison, John (1822). Murray, John (ed.). A System of Mechanical Philosophy. Vol. 4. London, England: Printed for J. Murray.
Maxwell, James Clerk, ed. (1967) [1879]. "Experiments on Electricity: Experimental determination of the law of electric force.". The Electrical Researches of the Honourable Henry Cavendish... (1st ed.). Cambridge, England: Cambridge University Press. pp. 104–113.
"Coulomb's law". Hyperphysics. Archived from the original on 2019-04-13. Retrieved 2007-11-30.
Feynman, Richard P. (1970). The Feynman Lectures on Physics Vol II. Addison-Wesley. ISBN 9780201021158.
Fitzpatrick, Richard (2006-02-02). "Coulomb's law". University of Texas. Archived from the original on 2015-07-09. Retrieved 2007-11-30.
"Charged rods". PhysicsLab.org. Archived from the original on 2014-10-10. Retrieved 2007-11-06.
Walker, Jearl; Halliday, David; Resnick, Robert (2014). Fundamentals of physics (10th ed.). Hoboken, NJ: Wiley. p. 614. ISBN 9781118230732. OCLC 950235056.
Le Système international d’unités PDF (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, p. 15, ISBN 978-92-822-2272-0
BIPM statement: Information for users about the proposed revision of the SI (PDF), archived from the original (PDF) on 2018-01-21, retrieved 2020-04-15
"Decision CIPM/105-13 (October 2016)". Archived from the original on 2017-08-24. Retrieved 2020-04-15. The day is the 144th anniversary of the Metre Convention.
Derived from k\text{e} = 1 / (4πε0) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Archived from the original on 2016-06-03. Retrieved 2019-05-20.
Discussion on physics teaching innovation: Taking Coulomb's law as an example. CRC Press. 2015-07-28. pp. 465–468. doi:10.1201/b18636-105. ISBN 978-0-429-22704-2. {{cite book}}: |work= ignored (help)
Griffiths, David J. (2013). Introduction to Electrodynamics (4th ed.). Prentice Hall. p. 50.
Rosser, W. G. V. (1968). Classical Electromagnetism via Relativity. pp. 29–42. doi:10.1007/978-1-4899-6559-2. ISBN 978-1-4899-6258-4. Archived from the original on 2022-10-09. Retrieved 2022-10-09.
Heaviside, Oliver (1894). Electromagnetic waves, the propagation of potential, and the electromagnetic effects of a moving charge. Archived from the original on 2022-10-09. Retrieved 2022-10-09.
Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, NJ: Prentice Hall. p. 517. ISBN 0-13-805326-X. OCLC 40251748.
Purcell, Edward (2011-09-22). Electricity and Magnetism. Cambridge University Press. doi:10.1017/cbo9781139005043. ISBN 978-1-107-01360-5. Archived from the original on 2023-12-30. Retrieved 2022-10-09.
Griffiths, David J. (16 August 2018). Introduction to quantum mechanics (Third ed.). Cambridge, United Kingdom. ISBN 978-1-107-18963-8.
"Quantum Field Theory I + II" (PDF). Institute for Theoretical Physics, Heidelberg University. Archived (PDF) from the original on 2021-05-04. Retrieved 2020-09-24.
Baym, Gordon (2018). Lectures on quantum mechanics. Boca Raton. ISBN 978-0-429-49926-5. OCLC 1028553174.
Gould, Robert J. (21 July 2020). Electromagnetic processes. Princeton, N.J. ISBN 978-0-691-21584-6. OCLC 1176566442.
Related reading
edit
Coulomb, Charles Augustin (1788) [1785]. "Premier mémoire sur l'électricité et le magnétisme". Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 569–577.
Coulomb, Charles Augustin (1788) [1785]. "Second mémoire sur l'électricité et le magnétisme". Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 578–611.
Coulomb, Charles Augustin (1788) [1785]. "Troisième mémoire sur l'électricité et le magnétisme". Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 612–638.
Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 978-0-13-805326-0.
Tamm, Igor E. (1979) [1976]. Fundamentals of the Theory of Electricity (9th ed.). Moscow: Mir. pp. 23–27.
Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and Engineers (6th ed.). New York: W. H. Freeman and Company. ISBN 978-0-7167-8964-2. LCCN 2007010418.
Young, Hugh D.; Freedman, Roger A. (2010). Sears and Zemansky's University Physics: With Modern Physics (13th ed.). Addison-Wesley (Pearson). ISBN 978-0-321-69686-1.
External links
edit
Wikimedia Commons has media related to Coulomb's law.
Coulomb's Law on Project PHYSNET
Electricity and the Atom Archived 2009-02-21 at the Wayback Machine—a chapter from an online textbook
A maze game for teaching Coulomb's law—a game created by the Molecular Workbench software
Electric Charges, Polarization, Electric Force, Coulomb's Law Walter Lewin, 8.02 Electricity and Magnetism, Spring 2002: Lecture 1 (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.
Last edited 9 days ago by 2600:1002:B177:28AF:ECBC:3CAC:7228:21B0
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From Wikipedia, the free encyclopedia
For the Latin word dubius, see List of Latin and Greek words commonly used in systematic names § D.
In this Dutch name, the surname is van den Hoef.
DVBBS
DVBBS in 2021; Alexandre (left), Christopher (right)
DVBBS in 2021; Alexandre (left), Christopher (right)
Background information
Origin Orangeville, Ontario, Canada
Los Angeles, California, U.S.
Genres
EDMhousedance-pop
Years active 2012–present
Labels
UniversalKanaryDoornSpinnin'Ultra
Members
Alexandre van den Hoef
Christopher van den Hoef
Dvbbs (stylized in all caps as DVBBS; pronounced "dubs") is a Canadian production duo formed in 2012, composed of brothers Christopher van den Hoef and Alexandre van den Hoef. The duo is known for popularising the usage of ghost producers among big producers in the scene. [1][2]
History
Career
Christopher and Alexandre Van den Hoef are natives of Orangeville, Ontario.[3] Their father is Dutch and migrated to Canada, and their mother is Greek Canadian. The two DJs lived in Athens, Greece for two years. They are currently based in Los Angeles, California,[3] as well as Toronto, Ontario, and are signed with Josh Herman at STRVCTVRE Artist Management.[4]
DVBBS spent two months recording their debut extended play, Initio, released in March 2012.[1][5] The EP features influences from a variety of genres, including house, electronic, dubstep and reggae.[6][7]
Following their 2012 debut EP, ‘Initio,’ DVBBS gained international acclaim via their chart-topping single, “Tsunami,” in 2013. As the duo’s first-ever certified hit, the platinum-selling “Tsunami” today counts over 1 billion streams. The 2014 vocal mix of the track, “Tsunami (Jump)” featuring English rapper Tinie Tempah, topped both the UK Singles and UK Dance charts, making DVBBS one of only 13 Canadian artists to ever chart at #1 on the Official UK Singles Chart.[8] DVBBS also count 12 platinum singles [9] via "Tinted Eyes", "West Coast", "IDWK", “Tsunami” and “Not Going Home,” the latter of which also received a Juno Awards nomination for Dance Recording of the Year in 2018.[10]
Their hit single came with "Tsunami". It had been promoted by Dutch DJ Sander van Doorn, although he denied being the producer.[11] The song was confirmed by Pete Tong to be the work of DVBBS and Borgeous when Tong played the song on his BBC Radio 1 show on August 16, 2013.[12] It was officially released on Sander Van Doorn's label Doorn Records on August 19, 2013.[11][13] It also reached #1 on the Beatport 100.[14] A vocal remix featuring British rapper Tinie Tempah entitled "Tsunami (Jump)" was released in Tempah's native United Kingdom in March 2014, on Ministry of Sound Recordings. DVBBS released their debut album, 'Sleep' on October 22, 2021.
Live performances
DVBBS live at Airbeat One Festival 2016 in Germany
As one of the top live acts in dance music—clocking in more than 750 shows in the past five years—they have performed across the world’s top festivals. DVBBS have performed with Tiësto, NERVO, Steve Aoki, Martin Garrix, Showtek, Sander Van Doorn, DJ Snake and others.[3][7] They have performed at most major music festivals.[15] The duo have been playing at mainstage Tomorrowland, Ultra Music Festival, and Creamfields, among many others.
Awards
2013: "One to Watch" award at the Canadian Urban Music Awards[1]
2014: "Dance recording of the year" Nominee - Juno Awards [16]
2014: "Track of the year" - EMPO Awards
2018: "Dance recording of the year" Nominee - Juno Awards [16]
2019: "Single of the year" Canadian Music Week - Indie Awards [17]
2021 "Electronic/ Dance Artist/ Group or Duo of the Year" Canadian Music Week - Indie Awards [18]
Discography
Studio albums
List of studio albums
Title Details
Sleep[19]
Released: October 22, 2021[20]
Label: Ultra, Kanary
Formats: Digital download
Extended plays
List of extended plays
Title EP details Peak chart positions
CAN
[21]
Initio
Released: March 20, 2012
Label: Universal Music Canada
Format: Digital download
—
We Were Young[22]
Released: December 22, 2014
Label: Kanary
Format: Digital download
—
Beautiful Disaster
Released: December 2, 2016
Label: Universal Music Canada
Format: Digital download
98
Blood of My Blood[23]
Released: October 20, 2017
Label: Kanary, Ultra
Format: Digital download
—
Nothing to See Here[24]
Released: August 28, 2020
Label: Ultra, Kanary
Formats: Digital download
—
"—" denotes an extended play that did not chart or was not released.
Singles
Title Year Peak chart positions Certifications Album
CAN
[25] AUT
[26] BEL
[27] DEN
[28] FRA
[29] GER
[30] NLD
[31] SWE
[32] SWI
[33] UK
"Love & Lies" 2012 — — — — — — — — — — Non-album singles
"We Are Electric"
(featuring Simon Wilcox) 2013 — — — — — — — — — —
"We Know"
(with Swanky Tunes and Eitro) — — — — — — — — — —
"Tsunami"
(with Borgeous) 80 14 1 8 3 14 1 3 17 —
MC: Platinum[34]
BEA: 2× Platinum[35]
GLF: 2× Platinum[36]
IFPI DEN: 2× Platinum[37]
"Stampede"
(with Borgeous and Dimitri Vegas & Like Mike) — — 30 — 129 — — — — — We Were Young
"Raveology"
(with Vinai) 2014 — — 84 — — — — — — — Non-album single
"Tsunami (Jump)"
(with Borgeous featuring Tinie Tempah) — — — — — — — — — 1
ARIA: Platinum[38]
BPI: Gold[39]
Demonstration
"Immortal"
(with Tony Junior) — — — — — — — — — — We Were Young
"We Were Young" — — — — — — — — — —
"Gold Skies"
(with Sander van Doorn and Martin Garrix featuring Aleesia) 57 — 57 — 183 — 94 — — 49 Gold Skies
"Deja Vu"
(with Joey Dale featuring Delora) — — — — — — — — — — We Were Young
"Pyramids"
(with Dropgun featuring Sanjin) — — 39 — — — — — — —
"Voodoo"
(with Jay Hardway) 2015 — — — — — — — — — — Non-album singles
"Always" — — — — — — — — — —
"White Clouds" — — — — — — — — — —
"Raveheart" — — — — — — — — — —
"Never Leave" — — 84 — — — — — — —
"Angel"
(featuring Dante Leon) 2016 — — — — — — — — — —
"La La Land"
(with Shaun Frank featuring Delaney Jane) 93 — — — 157 — 41 — — —
MC: Gold[34]
"Switch"
(with MOTi) — — — — — — — — — —
"24K" — — — — — — — — — — Beautiful Disaster
"Ur on My Mind" — — — — 82
[40] — — — — —
"Not Going Home"
(with CMC$ featuring Gia Koka) 52 — — — — — 83 — — —
MC: 2× Platinum[34]
"Without U"
(with Steve Aoki featuring 2 Chainz) 2017 — — — — — — — — — — Steve Aoki Presents Kolony
"You Found Me"
(featuring Belly) — — — — — — — — — — Blood of My Blood
"Parallel Lines"
(with CMC$ featuring Happy Sometimes) — — — — — — — — — —
MC: Gold[34]
"Cozee"
(featuring Cisco Adler) — — — — — — — — — —
"Make It Last"
(with Nervo) — — — — — — — — — —
"Good Time"
(featuring 24hrs) — — — — — — — — — —
"I Love It"
(with Cheat Codes)[41] 2018 — — — — — — — — — — Level 1
"Idwk"
(with Blackbear)[42] 67 — — — — — — — — —
MC: Platinum[34]
Non-album single
"Listen Closely"
(featuring Safe)[43] — — — — — — — — — — Nothing to See Here
"Somebody Like You"featuring Saro 2019 — — — — — — — — — —
"Gomf"
(featuring Bridge)[45] — — — — — — — — — —
"Need U"[46] — — — — — — — — — —
"Wrong About You"[47] 2020 — — — — — — — — — —
"Tinted Eyes"
(featuring Blackbear and 24kGoldn)[48] 62 — — — — — — — — —
MC: Gold[34]
Non-album single
"Swim"with Sondr featuring Keelan Donovan — — — — — — — — — — Nothing to See Here
"No Time"with Cheat Codes featuring Wiz Khalifa and Prince$$ Rosie — — — — — — — — — — Non-album singles
"West Coast"
(featuring Quinn XCII)[51] 67 — — — — — — — — —
MC: Gold[34]
"Too Much"with Dimitri Vegas & Like Mike and Roy Woods 2021 — — — — — — — — — —
"I Don't"with Johnny Orlando — — — — — — — — — —
"Fool for Ya"[54] — — — — — — — — — —
"Lose My Mind"[55] — — — — — — — — — — Sleep
"Losing Sleep"
(with Powfu)[56] — — — — — — — — — —
"Leave the World Behind"
(with GATTÜSO and Alida)[57] — — — — — — — — — —
"Victory"
(featuring YBN Nahmir)[58] — — — — — — — — — —
"Say It"
(with Space Primates featuring Gashi)[59] — — — — — — — — — —
"Set Me Free"
(featuring Aloe Blacc)[60] — — — — — — — — — —
"When the Lights Go Down"with Galantis featuring Cody Simpson 2022 — — — — — — — — — — Non-album singles
"Ride or Die"with Kideko and Haj — — — — — — — — — —
"Summer Nights"with Brandyn Burnette — — — — — — — — — —
"Love Till It's Over"featuring Mkla — — — — — — — — — —
"Ocean of Tears"with Imanbek — — — — — — — — — —
"Where Do We Go"with Lumix — — — — — — — — — —
"Just Words"[67] — — — — — — — — — —
"After Hours"[68] 2023 — — — — — — — — — —
"Sh Sh Sh (Hit That)"
(with Wiz Khalifa featuring Urfavxboyfriend and Goldsoul)[69] — — — — — — — — — —
"Next to You"
(with Loud Luxury featuring Kane Brown)[70] 34 — — — — — — — — —
MC: Gold[34]
"Synergy"with Timmy Trumpet — — — — — — — — — —
"Crew Thang"with Jeremih and Sk8 — — — — — — — — — —
"—" denotes a recording that did not chart or was not released in that territory.
DJ Magazine Top 100 DJ
Year Position Notes Ref.
2014 20
2015 16 Up 4 [73]
2016 24 Down 8
2017 22 Up 2
2018 16 Up 6
2019 23 Down 7
2020 55 Down 32
2021 n/a Out
References
Sean Plummer,“Toronto brothers Chris and Alex power DVBBS,” Archived 2013-06-25 at the Wayback Machine MSN Entertainment Canada, October 5, 2012.
“DUBBS Shocks the House,” Archived 2011-03-20 at the Wayback Machine MuchMusic, November 22, 2010.
Jordan Nunziato, “Orangeville natives DVBBS to headline town’s 150th celebration,” Orangeville Citizen, June 13, 2013.
"DVBBS' Manager Josh Herman Shares Six Tips on How to Become a Successful Music Producer". LA Weekly. February 12, 2020.
Bill Tremblay, “DVBBS headlines eclectic birthday bash,” Orangeville.com, June 27, 2013.
Cristina Almudevar, “Who is DVBBS?” The Cord Weekly, November 28, 2012.
Greg Belasco, “DVBBS – Initio,” Earmilk, March 13, 2012.
"DVBBS & BORGEOUS/TINIE TEMPAH | full Official Chart History | Official Charts Company". http://www.officialcharts.com.
"Gold/Platinum". Music Canada.
"DVBBS". Spotify.
Zel McCarthy (August 16, 2013). "Sander van Doorn's 'Tsunami' Mystery Almost Revealed". Billboard. Retrieved January 25, 2014.
“Avicii World Exclusive, Ray Foxx, Felix Da Housecat, DJ W!ld & Chris Malinchak,” BBC Radio 1, August 16, 2013.
“DVBBS & Borgeous – Tsunami (Preview),” Archived 2013-09-23 at the Wayback Machine DJ Magazine, August 16, 2013.
Sean Lewis, “DVBBS and Borgeous storm the Beatport 100 with ‘Tsunami’,” Archived 2013-09-26 at the Wayback Machine Beatport, August 26, 2013.
Katie Moran, “DVBBS & Borgeous hit us with ‘Tsunami’,” The DJ List, August 19, 2013.
"Past Nominees + Winners".
"DVBBS WIN SINGLE OF THE YEAR AT THE 2019 CANADIAN MUSIC WEEK INDIE AWARDS". Reservoir Media. May 13, 2019.
"INDIES Awards 2021 | INDIES".
@DVBBS (March 22, 2021). "our album is called 'SLEEP'" (Tweet). Retrieved March 27, 2021 – via Twitter.
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"DVBBS - Chart history (Canadian Albums)". Billboard. Retrieved December 13, 2016.
"We Were Young by DVBBS". December 20, 2014 – via music.apple.com.
Kat Bein (September 22, 2017). "R&B and Future-Pop Get Cozy on DVBBS' NERVO Collab 'Make It Last': Listen". Billboard.
"Nothing to See Here by DVBBS on Apple Music". Apple Music. Retrieved September 5, 2020.
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"Next to You": "Canadian Hot 100: May 20, 2023". Billboard. Retrieved May 16, 2023.
"Discographie DVBBS". Austrian Charts Portal. Hung Medien.
"Discografie DVBBS". Belgium (Flanders) Charts Portal. Hung Medien.
"Discography DVBBS". Danish Charts Portal. Hung Medien.
"Discographie DVBBS". French Charts Portal. Hung Medien.
"Discographie DVBBS". German Charts Portal. Hung Medien. Archived from the original on December 14, 2014.
"Discografie DVBBS". Dutch Charts Portal. Hung Medien.
"Discography DVBBS". Swedish Charts Portal. Hung Medien.
"Discographie DVBBS". Swiss Charts Portal. Hung Medien.
"Canadian certifications – Dvbbs". Music Canada. Retrieved August 22, 2023.
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"Danish single certifications – Dvbbs & Borgeous – Tsunami". IFPI Danmark. Retrieved August 22, 2023.
"ARIA Charts – Accreditations – 2014 Singles" (PDF). Australian Recording Industry Association. Retrieved August 22, 2023.
"British certifications – Dvbbs". British Phonographic Industry. Retrieved August 22, 2023. Type Dvbbs in the "Search BPI Awards" field and then press Enter.
"Le Top de la semaine : Top Singles Téléchargés - SNEP (Week 36, 2016)" (in French). Syndicat National de l'Édition Phonographique. Archived from the original on September 15, 2016. Retrieved September 10, 2016.
"I Love It – Single by Cheat Codes & DVBBS". iTunes Store (AU). 18 May 2018. Retrieved May 18, 2018.
"IDWK – Single by DVBBS & Blackbear". iTunes Store (CA). June 2018. Retrieved June 1, 2018.
"Listen Closely (feat. Safe) - Single by DVBBS on Apple Music". iTunes Store. 28 September 2018. Retrieved September 30, 2018.
Saponara, Michael (March 8, 2019). "DVBBS teams up With Saro for moody 'Somebody Like You': Premiere". Billboard. Archived from the original on March 14, 2019. Retrieved March 14, 2019.
Kassam, Alshaan (2019-05-06). "Free yourself this summer with DVBBS new single "GOMF" feat. BRIDGE". Earmilk. Retrieved 12 May 2019.
Bein, Kat (August 15, 2019). "Dvbbs chase summer highs on new track 'Need U': Exclusive". Billboard. Archived from the original on August 25, 2019. Retrieved August 31, 2019.
Karakolis, Konstantinos (March 21, 2020). "DVBBS change course with house single 'Wrong About You'". EDM.com. Archived from the original on March 24, 2020. Retrieved March 24, 2020.
"Tinted Eyes (feat. blackbear & 24kGoldn) - Single by DVBBS on Apple Music". Apple Music. Retrieved June 26, 2020.
"Swim (feat. Keelan Donovan) - Single by DVBBS & Sondr on Apple Music". Apple Music. Retrieved August 6, 2020.
"No Time (feat. PRINCE$$ ROSIE & Wiz Khalifa) - Single by Cheat Codes & DVBBS on Apple Music". Apple Music. Retrieved August 14, 2020.
"West Coast (feat. Quinn XCII) - Single by DVBBS on Apple Music". Apple Music. Retrieved October 16, 2020.
"Too Much - Single by Dimitri Vegas & Like Mike, DVBBS & Roy Woods on Apple Music". Apple Music. Retrieved February 18, 2021.
"I Don't - Single by Johnny Orlando & DVBBS on Apple Music". Apple Music. Retrieved March 27, 2021.
Astronaut, Dancing (April 27, 2021). "DVBBS debut first single from upcoming album 'Fool For Ya'". Dancing Astronaut. Archived from the original on April 27, 2021. Retrieved May 7, 2021.
"Lose My Mind - Single by DVBBS on Apple Music". Apple Music. Retrieved May 21, 2021.
"Losing Sleep - Single by DVBBS & Powfu on Apple Music". Apple Music. Retrieved June 25, 2021.
"Leave the World Behind - Single by DVBBS, GATTÜSO & Alida on Apple Music". Apple Music. Retrieved August 14, 2021.
"Victory (feat. YBN Nahmir) - Single by DVBBS on Apple Music". Apple Music. Retrieved September 12, 2021.
"Say It (feat. GASHI) by DVBBS, Space Primates & GASHI on Spotify". Spotify. 8 October 2021. Retrieved October 8, 2021.
Doole, Kerry (October 24, 2021). "DVBBS: Set Me Free-feat. Aloe Blacc". FYIMusicNews. Retrieved December 19, 2021.
"When the Lights Go Down (feat. Cody Simpson) - Single by DVBBS & Galantis". Apple Music. Retrieved 22 January 2022.
"Ride or Die - Single by DVBBS, Kideko, & Haj". Apple Music. Retrieved 31 July 2022.
"Summer Nights - Single by DVBBS & Brandyn Burnette". Apple Music. Retrieved 31 July 2022.
"Love Till It's Over (feat. MKLA) - Single by DVBBS". Apple Music. Retrieved 31 July 2022.
"IMANBEK & DVBBS Join Forces on New Single "Ocean of Tears"". EDM House Network. 26 August 2022. Retrieved 27 August 2022.
"Where Do We Go - Single by LUM!X & DVBBS". Apple Music. Retrieved November 1, 2022.
"Just Words - Single by DVBBS". Apple Music. Retrieved November 1, 2022.
"After Hours – Single by DVBBS on Apple Music". Apple Music. Retrieved February 26, 2023.
"SH SH SH (Hit That) [feat. Urfavxboyfriend & Goldsoul] – Single by DVBBS & Wiz Khalifa on Apple Music". Apple Music. Retrieved February 26, 2023.
"Next To You (feat. Kane Brown) – Single by Loud Luxury & DVBBS on Apple Music". Apple Music. Retrieved February 26, 2023.
"Synergy - Single by Timmy Trumpet & DVBBS". Apple Music. Retrieved June 17, 2023.
"Crew Thang - Single by DVBBS, Jeremih & SK8". Apple Music. Retrieved June 17, 2023.
"Poll 2014-2021: DVBBS". DJMag.com.
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