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
The basic relationship between the sine and cosine is given by the Pythagorean identity:
sin 2 θ + cos 2 θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
where sin 2 θ {\displaystyle \sin ^{2}\theta } means ( sin θ ) 2 {\displaystyle {(\sin \theta )}^{2}} and cos 2 θ {\displaystyle \cos ^{2}\theta } means ( cos θ ) 2 . {\displaystyle {(\cos \theta )}^{2}.}
This can be viewed as a version of the Pythagorean theorem, and follows from the equation x 2 + y 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 θ = ± 1 − cos 2 θ , cos θ = ± 1 − sin 2 θ . {\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 θ . {\displaystyle \theta .}
Dividing this identity by sin 2 θ {\displaystyle \sin ^{2}\theta } , cos 2 θ {\displaystyle \cos ^{2}\theta } , or both yields the following identities:
1 + cot 2 θ = csc 2 θ 1 + tan 2 θ = sec 2 θ sec 2 θ + csc 2 θ = sec 2 θ csc 2 θ {\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):
Reflections, shifts, and periodicity By examining the unit circle, one can establish the following properties of the trigonometric functions.
Reflections
When the direction of a Euclidean vector is represented by an angle θ , {\displaystyle \theta ,} this is the angle determined by the free vector (starting at the origin) and the positive x {\displaystyle x} -unit vector. The same concept may also be applied to lines in an Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x {\displaystyle x} -axis. If a line (vector) with direction θ {\displaystyle \theta } is reflected about a line with direction α , {\displaystyle \alpha ,} then the direction angle θ ′ {\displaystyle \theta ^{\prime }} of this reflected line (vector) has the value
θ ′ = 2 α − θ . {\displaystyle \theta ^{\prime }=2\alpha -\theta .}
The values of the trigonometric functions of these angles θ , θ ′ {\displaystyle \theta ,\;\theta ^{\prime }} for specific angles α {\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.
Shifts and periodicity
Signs The sign of trigonometric functions depends on quadrant of the angle. If − π < θ ≤ π {\displaystyle {-\pi }<\theta \leq \pi } and sgn is the sign function,
sgn ( sin θ ) = sgn ( csc θ ) = { + 1 if 0 < θ < π − 1 if − π < θ < 0 0 if θ ∈ { 0 , π } sgn ( cos θ ) = sgn ( sec θ ) = { + 1 if − π 2 < θ < π 2 − 1 if − π < θ < − π 2 or π 2 < θ < π 0 if θ ∈ { − π 2 , π 2 } sgn ( tan θ ) = sgn ( cot θ ) = { + 1 if − π < θ < − π 2 or 0 < θ < π 2 − 1 if − π 2 < θ < 0 or π 2 < θ < π 0 if θ ∈ { − π 2 , 0 , π 2 , π } {\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&{\text{if}}\ \ 0<\theta <\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <0\\0&{\text{if}}\ \ \theta \in \{0,\pi \}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&{\text{if}}\ \ {-{\tfrac {\pi }{2}}}<\theta <{\tfrac {\pi }{2}}\\-1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {\pi }{2}}\ \ {\text{or}}\ \ {\tfrac {\pi }{2}}<\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr \}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {\pi }{2}}\ \ {\text{or}}\ \ 0<\theta <{\tfrac {\pi }{2}}\\-1&{\text{if}}\ \ {-{\tfrac {\pi }{2}}}<\theta <0\ \ {\text{or}}\ \ {\tfrac {\pi }{2}}<\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {\pi }{2}}},0,{\tfrac {\pi }{2}},\pi {\bigr \}}\end{cases}}\end{aligned}}}
The trigonometric functions are periodic with common period 2 π , {\displaystyle 2\pi ,} so for values of θ outside the interval ( − π , π ] , {\displaystyle ({-\pi },\pi ],} they take repeating values (see § Shifts and periodicity above). The sign of a sinusoid or cosinusoid can be used to define a normalized square wave. For example, the functions sgn ( sin x ) {\displaystyle \operatorname {sgn}(\sin x)} and sgn ( cos x ) {\displaystyle \operatorname {sgn}(\cos x)} take values ±1 and correspond to square waves with a phase shift of π/2.
Angle sum and difference identities
These are also known as the angle addition and subtraction theorems (or formulae).
sin ( α + β ) = sin α cos β + cos α sin β sin ( α − β ) = sin α cos β − cos α sin β cos ( α + β ) = cos α cos β − sin α sin β cos ( α − β ) = cos α cos β + sin α sin β {\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 ( α − β ) {\displaystyle \sin(\alpha -\beta )} and cos ( α − β ) {\displaystyle \cos(\alpha -\beta )} can be derived from the angle sum versions (and vice versa) by substituting − β {\displaystyle -\beta } for β {\displaystyle \beta } and using the facts that sin ( − β ) = − sin ( β ) {\displaystyle \sin(-\beta )=-\sin(\beta )} and cos ( − β ) = cos ( β ) {\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. They can also be seen as expressing the dot product and cross product of two vectors in terms of the cosine and the sine of the angle between them. 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.
Sines and cosines of sums of infinitely many angles When the series ∑ i = 1 ∞ θ i {\textstyle \sum _{i=1}^{\infty }\theta _{i}} converges absolutely then
sin ( ∑ i = 1 ∞ θ i ) = ∑ odd k ≥ 1 ( − 1 ) k − 1 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ( ∏ i ∈ A sin θ i ∏ i ∉ A cos θ i ) cos ( ∑ i = 1 ∞ θ i ) = ∑ even k ≥ 0 ( − 1 ) k 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ( ∏ i ∈ A sin θ i ∏ i ∉ A cos θ i ) . {\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 ∑ i = 1 ∞ θ i {\textstyle \sum _{i=1}^{\infty }\theta _{i}} converges absolutely, it is necessarily the case that lim i → ∞ θ i = 0 , {\textstyle \lim _{i\to \infty }\theta _{i}=0,} lim i → ∞ sin θ i = 0 , {\textstyle \lim _{i\to \infty }\sin \theta _{i}=0,} and lim i → ∞ cos θ i = 1. {\textstyle \lim _{i\to \infty }\cos \theta _{i}=1.} Particularly, 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 θ i {\displaystyle \theta _{i}} 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 Let e k {\displaystyle e_{k}} (for k = 0 , 1 , 2 , 3 , … {\displaystyle k=0,1,2,3,\ldots } ) be the kth-degree elementary symmetric polynomial in the variables
x i = tan θ i {\displaystyle x_{i}=\tan \theta _{i}}
for i = 0 , 1 , 2 , 3 , … , {\displaystyle i=0,1,2,3,\ldots ,} that is,
e 0 = 1 e 1 = ∑ i x i = ∑ i tan θ i e 2 = ∑ i < j x i x j = ∑ i < j tan θ i tan θ j e 3 = ∑ i < j < k x i x j x k = ∑ i < j < k tan θ i tan θ j tan θ k ⋮ ⋮ {\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&\ \ \vdots &&\ \ \vdots \end{aligned}}}
Then
tan ( ∑ i θ i ) =