List of trigonometric identities explained

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

See main article: Pythagorean trigonometric identity.

The basic relationship between the sine and cosine is given by the Pythagorean identity:

$$\backslash sin^2\backslash theta\; +\; \backslash cos^2\backslash theta\; =\; 1,$$

where

\sin ^{2} \theta means (\sin \theta) ^{2} and \cos ^{2} \theta means (\cos \theta) ^{2.}

This can be viewed as a version of the Pythagorean theorem, and follows from the equation

x ^{2} + y ^{2} = 1 for the unit circle. This equation can be solved for either the sine or the cosine:

where the sign depends on the quadrant of

\theta.

Dividing this identity by

\sin ^{2} \theta , \cos ^{2} \theta , or both yields the following identities:

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

in terms of! scope="col"

\sin \theta	\csc \theta	\cos \theta	\sec \theta	\tan \theta	\cot \theta
scope=row	\sin \theta =	\sin \theta	(1)/(\csc \theta)	\pm\sqrt{ 1 - \cos ^{2} \theta}	\pm (\sqrt{\sec ^{2} \theta -1)/() }	\pm (\tan \theta)/(\sqrt{ 1+ \tan ^{2} \theta) }	\pm (1)/(\sqrt{ 1+ \cot ^{2} \theta) }
scope=row	\csc \theta =	(1)/(\sin \theta)	\csc \theta	\pm (1)/(\sqrt{ 1- \cos ^{2} \theta) }	\pm (\sec \theta)/(\sqrt{\sec ^{2} \theta -1) }	\pm (\sqrt{ 1+ \tan ^{2} \theta)/() }	\pm\sqrt{ 1 + \cot ^{2} \theta}
scope=row	\cos \theta =	\pm\sqrt{ 1 - \sin ^{2\theta}}	\pm (\sqrt{\csc ^{2} \theta -1)/() }	\cos \theta	(1)/(\sec \theta)	\pm (1)/(\sqrt{ 1+ \tan ^{2} \theta) }	\pm (\cot \theta)/(\sqrt{ 1+ \cot ^{2} \theta) }
scope=row	\sec \theta =	\pm (1)/(\sqrt{ 1- \sin ^{2} \theta) }	\pm (\csc \theta)/(\sqrt{\csc ^{2} \theta -1) }	(1)/(\cos \theta)	\sec \theta	\pm\sqrt{ 1 + \tan ^{2} \theta}	\pm (\sqrt{ 1+ \cot ^{2} \theta)/() }
scope=row	\tan \theta =	\pm (\sin \theta)/(\sqrt{ 1- \sin ^{2} \theta) }	\pm (1)/(\sqrt{\csc ^{2} \theta -1) }	\pm (\sqrt{ 1- \cos ^{2} \theta)/() }	\pm\sqrt{\sec ^{2} \theta - 1 }	\tan \theta	(1)/(\cot \theta)
scope=row	\cot \theta =	\pm (\sqrt{ 1- \sin ^{2} \theta)/() }	\pm\sqrt{\csc ^{2} \theta - 1 }	\pm (\cos \theta)/(\sqrt{ 1- \cos ^{2} \theta) }	\pm (1)/(\sqrt{\sec ^{2} \theta -1) }	(1)/(\tan \theta)	\cot \theta

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

\theta, this is the angle determined by the free vector (starting at the origin) and the positive 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 x -axis. If a line (vector) with direction \theta is reflected about a line with direction \alpha, then the direction angle \theta ^{\prime} of this reflected line (vector) has the value$$\backslash theta^\; =\; 2\; \backslash alpha\; -\; \backslash theta.$$

The values of the trigonometric functions of these angles

\theta, \theta ^{\prime} for specific angles \alpha satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as .

\theta reflected in \alpha = 0 ^[1] odd/even identities	\theta reflected in \alpha = (\pi)/(4)	\theta reflected in \alpha = (\pi)/(2)	\theta reflected in \alpha = (3 \pi)/(4)	\theta reflected in \alpha = \pi compare to \alpha = 0
\sin( - \theta) = - \sin \theta	\sin\left(\tfrac{\pi}{ 2 } - \theta\right) = \cos \theta	\sin(\pi - \theta) = + \sin \theta	\sin\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) =- \cos \theta	\sin( 2 \pi - \theta) = - \sin(\theta) = \sin( - \theta)
\cos( - \theta) =+ \cos \theta	\cos\left(\tfrac{\pi}{ 2 } - \theta\right) = \sin \theta	\cos(\pi - \theta) = - \cos \theta	\cos\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) = - \sin \theta	\cos( 2 \pi - \theta) = + \cos(\theta) = \cos( - \theta)
\tan( - \theta) = - \tan \theta	\tan\left(\tfrac{\pi}{ 2 } - \theta\right) = \cot \theta	\tan(\pi - \theta) = - \tan \theta	\tan\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) = + \cot \theta	\tan( 2 \pi - \theta) = - \tan(\theta) = \tan( - \theta)
\csc( - \theta) = - \csc \theta	\csc\left(\tfrac{\pi}{ 2 } - \theta\right) = \sec \theta	\csc(\pi - \theta) =+ \csc \theta	\csc\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) = - \sec \theta	\csc( 2 \pi - \theta) = - \csc(\theta) = \csc( - \theta)
\sec( - \theta) = + \sec \theta	\sec\left(\tfrac{\pi}{ 2 } - \theta\right) = \csc \theta	\sec(\pi - \theta) = - \sec \theta	\sec\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) = - \csc \theta	\sec( 2 \pi - \theta) = + \sec(\theta) = \sec( - \theta)
\cot( - \theta) = - \cot \theta	\cot\left(\tfrac{\pi}{ 2 } - \theta\right) = \tan \theta	\cot(\pi - \theta) = - \cot \theta	\cot\left(\tfrac{ 3 \pi}{ 2 } - \theta\right) = + \tan \theta	\cot( 2 \pi - \theta) = - \cot(\theta) = \cot( - \theta)

Shifts and periodicity

Shift by one quarter period	Shift by one half period	Shift by full periods^[2]	Period
\sin(\theta \pm \tfrac{\pi}{ 2 }) = \pm\cos \theta	\sin(\theta + \pi) = - \sin \theta	\sin(\theta + k ⋅ 2 \pi) = + \sin \theta	2 \pi
\cos(\theta \pm \tfrac{\pi}{ 2 }) = \mp\sin \theta	\cos(\theta + \pi) = - \cos \theta	\cos(\theta + k ⋅ 2 \pi) = + \cos \theta	2 \pi
\csc(\theta \pm \tfrac{\pi}{ 2 }) = \pm\sec \theta	\csc(\theta + \pi) = - \csc \theta	\csc(\theta + k ⋅ 2 \pi) = + \csc \theta	2 \pi
\sec(\theta \pm \tfrac{\pi}{ 2 }) = \mp\csc \theta	\sec(\theta + \pi) = - \sec \theta	\sec(\theta + k ⋅ 2 \pi) = + \sec \theta	2 \pi
\tan(\theta \pm \tfrac{\pi}{ 4 }) = \tfrac{\tan \theta \pm 1 }{ 1 \mp \tan \theta}	\tan(\theta + \tfrac{\pi}{ 2 }) = - \cot \theta	\tan(\theta + k ⋅ \pi) = + \tan \theta	\pi
\cot(\theta \pm \tfrac{\pi}{ 4 }) = \tfrac{\cot \theta \mp 1 }{ 1 \pm \cot \theta}	\cot(\theta + \tfrac{\pi}{ 2 }) = - \tan\theta	\cot(\theta + k ⋅ \pi) = + \cot \theta	\pi

Signs

The sign of trigonometric functions depends on quadrant of the angle. If

{ - \pi} < \theta \leq \pi and is the sign function,

The trigonometric functions are periodic with common period

2 \pi, so for values of outside the interval ({ - \pi}, \pi], they take repeating values (see above).

Angle sum and difference identities

These are also known as the (or).

The angle difference identities for

\sin(\alpha - \beta) and \cos(\alpha - \beta) can be derived from the angle sum versions by substituting - \beta for \beta and using the facts that \sin( - \beta) = - \sin(\beta) and \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(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\sin \alpha \cos \beta \pm \cos \alpha \sin \beta ^[3]
Cosine	\cos(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\cos \alpha \cos \beta \mp \sin \alpha \sin \beta ^[4]
Tangent	\tan(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	(\tan \alpha \pm \tan \beta)/(1 \mp \tan \alpha \tan \beta) ^[5]
Cosecant	\csc(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	(\sec \alpha \sec \beta \csc \alpha \csc \beta)/(\sec \alpha \csc \beta \pm \csc \alpha \sec \beta) ^[6]
Secant	\sec(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	(\sec \alpha \sec \beta \csc \alpha \csc \beta)/(\csc \alpha \csc \beta \mp \sec \alpha \sec \beta)
Cotangent	\cot(\alpha \pm \beta)			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	(\cot \alpha \cot \beta \mp 1)/(\cot \beta \pm \cot \alpha) ^[7]
Arcsine	\arcsin x \pm \arcsin y			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\arcsin\left(x\sqrt{ 1- y ^{2}} \pm y\sqrt{ 1- x ^{2\vphantom{y}}\right)} ^[8]
Arccosine	\arccos x \pm \arccos y			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\arccos\left(xy \mp \sqrt{\left( 1- x ^{2\right)\left(1-y} ^{2\right)}\right)} ^[9]
Arctangent	\arctan x \pm \arctan y			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\arctan\left( (x \pm y)/(1 \mp xy) \right) ^[10]
Arccotangent	\arccot x \pm \arccot y			style='border-style: solid none solid none; text-align: center;'	=	style='border-style: solid solid solid none; text-align: left;'	\arccot\left( (xy \mp 1)/(y \pm x) \right)

Sines and cosines of sums of infinitely many angles

When the series $\backslash sum\_^\backslash infty\; \backslash theta\_i$ converges absolutely then

Because the series $\backslash sum\_^\backslash infty\; \backslash theta\_i$ converges absolutely, it is necessarily the case that $\backslash lim\_\; \backslash theta\_i\; =\; 0,$ $\backslash lim\_\; \backslash sin\; \backslash theta\_i\; =\; 0,$ and $\backslash lim\_\; \backslash cos\; \backslash theta\_i\; =\; 1.$ 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

\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} (for k = 0 , 1 , 2 , 3 , \ldots ) be the th-degree elementary symmetric polynomial in the variables$$x\_i\; =\; \backslash tan\; \backslash theta\_i$$for i = 0 , 1 , 2 , 3 , \ldots, that is,

Then

\Bigl(\sum_i \theta_i\Bigr)&= \frac\end

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:

and so on. The case of only finitely many terms can be proved by mathematical induction.^[11] The case of infinitely many terms can be proved by using some elementary inequalities.^[12]

Secants and cosecants of sums

where

e _{k} is the th-degree elementary symmetric polynomial in the variables x _{i} = \tan \theta _{i,} i = 1 , \ldots, n, 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.^[13] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

Ptolemy's theorem

See main article: Ptolemy's theorem.

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

ABCD , 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.^[14] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

By Thales's theorem,

\angle DAB and \angle DCB are both right angles. The right-angled triangles DAB and DCB both share the hypotenuse \overline{BD} of length 1. Thus, the side \overline{AB} = \sin \alpha , \overline{AD} = \cos \alpha , \overline{BC} = \sin \beta and \overline{CD} = \cos \beta .

By the inscribed angle theorem, the central angle subtended by the chord

\overline{AC} at the circle's center is twice the angle \angle ADC , i.e. 2 (\alpha + \beta) . Therefore, the symmetrical pair of red triangles each has the angle \alpha + \beta at the center. Each of these triangles has a hypotenuse of length $\backslash frac$, so the length of \overline{AC} is $2\; \backslash times\; \backslash frac\; \backslash sin(\backslash alpha\; +\; \backslash beta)$, i.e. simply \sin(\alpha + \beta) . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also \sin(\alpha + \beta) .

When these values are substituted into the statement of Ptolemy's theorem that

|\overline{AC}| ⋅ |\overline{BD}| = |\overline{AB}| ⋅ |\overline{CD}| + |\overline{AD}| ⋅ |\overline{BC}| , this yields the angle sum trigonometric identity for sine: \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta . The angle difference formula for \sin(\alpha - \beta) can be similarly derived by letting the side \overline{CD} serve as a diameter instead of \overline{BD} .^[14]

Multiple-angle and half-angle formulae

is the th Chebyshev polynomial	\cos (n\theta) = T _{n} (\cos \theta )
de Moivre's formula, is the imaginary unit	\cos (n\theta) + i\sin (n\theta) = (\cos \theta + i\sin \theta) ^{n} ^[15]

Multiple-angle formulae

Double-angle formulae

Formulae for twice an angle.

\sin ( 2 \theta) = 2 \sin \theta \cos \theta = (\sin \theta + \cos \theta) ^{2} - 1 = (2 \tan \theta)/(1+ \tan ^{2} \theta) \cos ( 2 \theta) = \cos ^{2} \theta - \sin ^{2} \theta = 2 \cos ^{2} \theta - 1 = 1 - 2 \sin ^{2} \theta = (1- \tan ^{2} \theta)/(1+ \tan ^{2} \theta) \tan ( 2 \theta) = (2 \tan \theta)/(1- \tan ^{2} \theta) \cot ( 2 \theta) = (\cot ^{2} \theta -1)/(2 \cot \theta) = (1- \tan ^{2} \theta)/(2 \tan \theta) \sec ( 2 \theta) = (\sec ^{2} \theta)/(2- \sec ^{2} \theta) = (1+ \tan ^{2} \theta)/(1- \tan ^{2} \theta) \csc ( 2 \theta) = (\sec \theta \csc \theta)/(2) = (1+ \tan ^{2} \theta)/(2 \tan \theta)

Triple-angle formulae

Formulae for triple angles.

\sin ( 3 \theta) =3 \sin\theta - 4 \sin ^{3\theta} =4 \sin\theta\sin\left( (\pi)/(- \theta\right)\sin\left( 3) (\pi)/(3) + \theta\right) \cos ( 3 \theta) = 4 \cos ^{3\theta} - 3 \cos\theta =4 \cos\theta\cos\left( (\pi)/(- \theta\right)\cos\left( 3) (\pi)/(3) + \theta\right) \tan ( 3 \theta) = (3 \tan\theta - \tan ^{3\theta})/(1-3 \tan ^{2\theta}) = \tan \theta\tan\left( (\pi)/(3) - \theta\right)\tan\left( (\pi)/(3) + \theta\right) \cot ( 3 \theta) = (3 \cot\theta - \cot ^{3\theta})/(1-3 \cot ^{2\theta}) \sec ( 3 \theta) = (\sec ^{3\theta})/(4-3 \sec ^{2\theta}) \csc ( 3 \theta) = (\csc ^{3\theta})/(3 \csc ^{2\theta-4})

Multiple-angle formulae

Formulae for multiple angles.^[16]

\begin{align} \sin(n\theta) & = \sum _{kodd} ( -1 ) ^{(k -1)/(2)} {n \choose k}\cos ^{n-k} \theta \sin ^{k} \theta = \sin\theta\sum ^{(n +1 )/ 2} _{i =0} \sum ^{i} _{j =0} ( -1 ) ^{i-j} {n \choose 2 i + 1 }{i \choose j} \cos ^{n-2(i-j)-1} \theta \\ {}& =2 ^{(n-1)} \prod ^{n -1} _{k =0} \sin(k\pi/n + \theta) \end{align} \cos(n\theta) = \sum _{keven} ( -1 ) ^{(k)/(2)} {n \choose k}\cos ^{n-k} \theta \sin ^{k} \theta = \sum ^{n/ 2} _{i =0} \sum ^{i} _{j =0} ( -1 ) ^{i-j} {n \choose 2 i}{i \choose j} \cos ^{n-2(i-j)} \theta \cos(( 2 n +1 )\theta) = ( -1 ) ^{n} 2 ^{2n} \prod ^{2 n} _{k =0} \cos(k\pi/( 2 n +1 ) - \theta) \cos( 2 n \theta) = ( -1 ) ^{n} 2 ^{2n-1} \prod ^{2 n -1} _{k =0} \cos(( 1+2 k)\pi/( 4 n) - \theta) \tan(n\theta) = (\sum _{k odd} ( -1 ) ^{(k -1)/(2)} {n \choose k)/(\tan) ^{k} \theta}{\sum _{keven} ( -1 ) ^{(k)/(2)} {n \choose k}\tan ^{k} \theta}

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the th multiple angle formula knowing the

(n -1 ) th and (n -2 ) th values.^[17] \cos(nx) can be computed from \cos((n -1 )x) , \cos((n -2 )x) , and \cos(x) with

$$\backslash cos(nx)=2\; \backslash cos\; x\; \backslash cos((n-1)x)\; -\; \backslash cos((n-2)x).$$

This can be proved by adding together the formulae

It follows by induction that

\cos(nx) is a polynomial of \cos x, the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly,

\sin(nx) can be computed from \sin((n -1 )x), \sin((n -2 )x), and \cos x with$$\backslash sin(nx)=2\; \backslash cos\; x\; \backslash sin((n-1)x)-\backslash sin((n-2)x)$$This can be proved by adding formulae for \sin((n -1 )x + x) and \sin((n -1 )x - x).

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

$$\backslash tan\; (nx)\; =\; \backslash frac\backslash ,.$$

Half-angle formulae

\cos \frac &= \sgn\left(\cos\frac\theta2\right) \sqrt \\[3pt]

\tan \frac&= \frac= \frac= \csc \theta - \cot \theta= \frac \\[6mu]

&= \sgn(\sin \theta) \sqrt\frac= \frac \\[3pt]

\cot \frac&= \frac= \frac= \csc \theta + \cot \theta= \sgn(\sin \theta) \sqrt\frac \\

\sec \frac&= \sgn\left(\cos\frac\theta2\right) \sqrt \\

\csc \frac&= \sgn\left(\sin\frac\theta2\right) \sqrt \\

\end

Also

Table

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^[18]	\begin{align} \sin ( 2 \theta) & = 2 \sin \theta \cos \theta \\ & = (2 \tan \theta)/(1+ \tan ^{2} \theta) \end{align}	\begin{align} \cos ( 2 \theta) & = \cos ^{2} \theta - \sin ^{2} \theta \\ & = 2 \cos ^{2} \theta - 1 \ & = 1 - 2 \sin ^{2} \theta \\ & = (1- \tan ^{2} \theta)/(1+ \tan ^{2} \theta) \end{align}	\tan ( 2 \theta) = (2 \tan \theta)/(1- \tan ^{2} \theta)	\cot ( 2 \theta) = (\cot ^{2} \theta -1)/(2 \cot \theta)
Triple-angle formula^[19]	\begin{align} \sin ( 3 \theta) & = - \sin ^{3\theta} + 3 \cos ^{2\theta} \sin\theta\\ & = - 4 \sin ^{3\theta} + 3 \sin\theta \end{align}	\begin{align} \cos ( 3 \theta) & = \cos ^{3\theta} - 3 \sin ^{2} \theta\cos \theta \\ & = 4 \cos ^{3\theta} - 3 \cos\theta \end{align}	\tan ( 3 \theta) = (3 \tan\theta - \tan ^{3\theta})/(1-3 \tan ^{2\theta})	\cot ( 3 \theta) = (3 \cot\theta - \cot ^{3\theta})/(1-3 \cot ^{2\theta})
Half-angle formula	\begin{align} &\sin (\theta)/(2) =sgn \left(\sin (\theta 2 \right))/(\sqrt{ (1- \cos \theta)/(2)) } \\ \\ &\left(\text\sin^2\frac = \frac\right)\end	\begin{align} &\cos (\theta)/(2) =sgn \left(\cos (\theta 2 \right))/(\sqrt{ (1+ \cos\theta)/(2)) } \\ \\ &\left(\text \cos^2\frac = \frac\right)\end	\begin{align} \tan (\theta)/(2) & = \csc \theta - \cot \theta \\ & = \pm \sqrt (1- \cos \theta)/(1+ \cos \theta) \\[ 3 pt] & = (\sin \theta)/(1+ \cos \theta) \\[ 3 pt] & = (1- \cos \theta)/(\sin \theta) \\[ 5 pt] \tan (η+ \theta)/(2) & = (\sin η+ \sin\theta)/(\cos η+ \cos\theta) \\[ 5 pt] \tan\left( (\theta)/(2) + (\pi)/(4) \right) & = \sec\theta + \tan\theta \\[ 5 pt] \sqrt{ (1- \sin\theta)/(1+ \sin\theta) } &= \frac \\[5pt] \tan\frac &= \frac \\ &\text \theta \in \left(-\tfrac,\tfrac \right)\end	\begin{align} \cot (\theta)/(2) & = \csc \theta + \cot \theta \\ & = \pm \sqrt (1+ \cos \theta)/(1- \cos \theta) \\[ 3 pt] & = (\sin \theta)/(1- \cos \theta) \\[ 4 pt] & = (1+ \cos \theta)/(\sin \theta) \end{align}

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.

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation, where

x is the value of the cosine function at the one-third angle and 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 are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Power-reduction formulae

Obtained by solving the second and third versions of the cosine double-angle formula.

Sine	Cosine	Other
\sin ^{2\theta} = (1- \cos ( 2 \theta))/(2)	\cos ^{2\theta} = (1+ \cos ( 2 \theta))/(2)	\sin ^{2\theta} \cos ^{2\theta} = (1- \cos ( 4 \theta))/(8)
\sin ^{3\theta} = (3 \sin\theta - \sin ( 3 \theta))/(4)	\cos ^{3\theta} = (3 \cos\theta + \cos ( 3 \theta))/(4)	\sin ^{3\theta} \cos ^{3\theta} = (3 \sin ( 2 \theta) - \sin ( 6 \theta))/(32)
\sin ^{4\theta} = (3-4 \cos ( 2 \theta) + \cos ( 4 \theta))/(8)	\cos ^{4\theta} = (3+4 \cos ( 2 \theta) + \cos ( 4 \theta))/(8)	\sin ^{4\theta} \cos ^{4\theta} = (3-4 \cos ( 4 \theta) + \cos ( 8 \theta))/(128)
\sin ^{5\theta} = (10 \sin\theta -5 \sin ( 3 \theta) + \sin ( 5 \theta))/(16)	\cos ^{5\theta} = (10 \cos\theta +5 \cos ( 3 \theta) + \cos ( 5 \theta))/(16)	\sin ^{5\theta} \cos ^{5\theta} = (10 \sin ( 2 \theta) -5 \sin ( 6 \theta) + \sin ( 10 \theta))/(512)

In general terms of powers of

\sin \theta or \cos \theta the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem.

if n is ...	\cos ^{n} \theta	\sin ^{n} \theta
n is odd	\cos ^{n\theta} = (2)/(2 ^{n}) \sum ^{(n -1)/(2)} _{k =0} \binom{n}{k} \cos{ ( (n -2 k)\theta ) }	\sin ^{n\theta} = (2)/(2 ^{n}) \sum ^{(n -1)/(2)} _{k =0} ( -1 ) ^{\left( (n -1)/(- k\right) 2)} \binom{n}{k} \sin{ ( (n -2 k)\theta ) }
n is even	\cos ^{n\theta} = (1)/(\binom{n}{ 2 ^{n}) (n)/(2) } + \frac \sum_^ \binom \cos	\sin ^{n\theta} = (1)/(\binom{n}{ 2 ^{n}) (n)/(2) } + \frac \sum_^ (-1)^ \binom \cos

Product-to-sum and sum-to-product identities

The product-to-sum identities^[20] 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.^[21] 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

\cos \theta \cos \varphi = {\cos(\theta - \varphi) + \cos(\theta + \varphi) \over 2 } \sin \theta \sin \varphi = {\cos(\theta - \varphi) - \cos(\theta + \varphi) \over 2 } \sin \theta \cos \varphi = {\sin(\theta + \varphi) + \sin(\theta - \varphi) \over 2 } \cos \theta \sin \varphi = {\sin(\theta + \varphi) - \sin(\theta - \varphi) \over 2 } \tan \theta \tan \varphi = (\cos(\theta - \varphi) - \cos(\theta + \varphi))/(\cos(\theta - \varphi) + \cos(\theta + \varphi)) \tan \theta \cot \varphi = (\sin(\theta + \varphi) + \sin(\theta - \varphi))/(\sin(\theta + \varphi) - \sin(\theta - \varphi)) \begin{align} \prod ^{n} _{k =1} \cos \theta _{k} & = (1)/(2 ^{n}) \sum _{e\in} \cos(e _{1\theta} _{1+ … +e} _{n\theta} _{n)} \\[ 6 pt] & where e = (e _{1,\ldots,e} _{n)} \in S = \{ 1 , -1 \} ^{n \end{align}} \prod ^{n} _{k =1} \sin\theta _{k = (( -1 ) ^{\left\lfloor (n)/(\right\rfloor 2)})/(2 ^{n})} \begin{cases} \displaystyle\sum _{e\in} \cos(e _{1\theta} _{1+ … +e} _{n\theta} _{n)\prod} ^{n} ^{j =1} e _{j} if n iseven ,\\ \displaystyle\sum _{e\in} \sin(e _{1\theta} _{1+ … +e} _{n\theta} _{n)\prod} ^{n} ^{j =1} e _{j} if n isodd \end{cases}

Sum-to-product identities

The sum-to-product identities are as follows:^[22]

\sin \theta \pm \sin \varphi = 2 \sin\left( (\theta \pm \varphi)/(2) \right) \cos\left( (\theta \mp \varphi)/(2) \right) \cos \theta + \cos \varphi = 2 \cos\left( (\theta + \varphi)/(2) \right) \cos\left( (\theta - \varphi)/(2) \right) \cos \theta - \cos \varphi = -2 \sin\left( (\theta + \varphi)/(2) \right) \sin\left( (\theta - \varphi)/(2) \right) \tan\theta\pm\tan\varphi = (\sin(\theta\pm \varphi))/(\cos\theta\cos\varphi)

Hermite's cotangent identity

See main article: Hermite's cotangent identity.

Charles Hermite demonstrated the following identity.^[23] Suppose

a _{1,} \ldots, a _{n} are complex numbers, no two of which differ by an integer multiple of . Let

$$A\_\; =\; \backslash prod\_\; \backslash cot(a\_k\; -\; a\_j)$$

(in particular,

A _{1,1} , being an empty product, is 1). Then

$$\backslash cot(z\; -\; a\_1)\backslash cdots\backslash cot(z\; -\; a\_n)\; =\; \backslash cos\backslash frac\; +\; \backslash sum\_^n\; A\_\; \backslash cot(z\; -\; a\_k).$$

The simplest non-trivial example is the case :

$$\backslash cot(z\; -\; a\_1)\backslash cot(z\; -\; a\_2)\; =\; -1\; +\; \backslash cot(a\_1\; -\; a\_2)\backslash cot(z\; -\; a\_1)\; +\; \backslash cot(a\_2\; -\; a\_1)\backslash cot(z\; -\; a\_2).$$

Finite products of trigonometric functions

For coprime integers,

$$\backslash prod\_^n\; \backslash left(2a\; +\; 2\backslash cos\backslash left(\backslash frac\; +\; x\backslash right)\backslash right)\; =\; 2\backslash left(T\_n(a)+^\backslash cos(n\; x)\; \backslash right)$$

where is the Chebyshev polynomial.

The following relationship holds for the sine function

$$\backslash prod\_^\; \backslash sin\backslash left(\backslash frac\backslash right)\; =\; \backslash frac.$$

More generally for an integer ^[24]

$$\backslash sin(nx)\; =\; 2^\backslash prod\_^\; \backslash sin\backslash left(\backslash frac\backslash pi\; +\; x\backslash right)\; =\; 2^\backslash prod\_^\; \backslash sin\backslash left(\backslash frac\backslash pi\; -\; x\backslash right).$$

or written in terms of the chord function $\backslash operatornamex\; \backslash equiv\; 2\backslash sin\backslash tfrac12x$,

$$\backslash operatorname(nx)\; =\; \backslash prod\_^\; \backslash operatorname\backslash left(\backslash frac2\backslash pi\; -\; x\backslash right).$$

This comes from the factorization of the polynomial $z^n\; -\; 1$ into linear factors (cf. root of unity): For a point on the complex unit circle and an integer,

$$z^n\; -\; 1\; =\; \backslash prod\_^\; z\; -\; \backslash exp\backslash Bigl(\backslash frac2\backslash pi\; i\backslash Bigr).$$

Linear combinations

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 and unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of

c and \varphi .

Sine and cosine

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,^[25]

$$a\backslash cos\; x+b\backslash sin\; x=c\backslash cos(x+\backslash varphi)$$

where

c and \varphi are defined as so:

given that

a ≠ 0 .

Arbitrary phase shift

More generally, for arbitrary phase shifts, we have

$$a\; \backslash sin(x\; +\; \backslash theta\_a)\; +\; b\; \backslash sin(x\; +\; \backslash theta\_b)=\; c\; \backslash sin(x+\backslash varphi)$$

where

c and \varphi satisfy:

More than two sinusoids

The general case reads$$\backslash sum\_i\; a\_i\; \backslash sin(x\; +\; \backslash theta\_i)\; =\; a\; \backslash sin(x\; +\; \backslash theta),$$where$$a^2\; =\; \backslash sum\_a\_i\; a\_j\; \backslash cos(\backslash theta\_i\; -\; \backslash theta\_j)$$and$$\backslash tan\backslash theta\; =\; \backslash frac.$$

Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange, are:^{[26] [27] [28]} for

\theta \not\equiv 0 \pmod{ 2 \pi}.

A related function is the Dirichlet kernel:

$$D\_n(\backslash theta)\; =\; 1\; +\; 2\backslash sum\_^n\; \backslash cos\; k\backslash theta=\; \backslash frac.$$

A similar identity is^[29]

$$\backslash sum\_^n\; \backslash cos\; (2k\; -1)\backslash alpha\; =\; \backslash frac.$$

The proof is the following. By using the angle sum and difference identities,$$\backslash sin\; (A\; +\; B)\; -\; \backslash sin\; (A\; -\; B)\; =\; 2\; \backslash cos\; A\; \backslash sin\; B.$$Then let's examine the following formula,

$$2\; \backslash sin\; \backslash alpha\; \backslash sum\_^n\; \backslash cos\; (2k\; -\; 1)\backslash alpha\; =\; 2\backslash sin\; \backslash alpha\; \backslash cos\; \backslash alpha\; +\; 2\; \backslash sin\; \backslash alpha\; \backslash cos\; 3\backslash alpha+\; 2\; \backslash sin\; \backslash alpha\; \backslash cos\; 5\; \backslash alpha\; +\; \backslash ldots\; +\; 2\; \backslash sin\; \backslash alpha\; \backslash cos\; (2n\; -\; 1)\; \backslash alpha$$and this formula can be written by using the above identity,

So, dividing this formula with

2 \sin \alpha completes the proof.

Certain linear fractional transformations

If

f(x) is given by the linear fractional transformation$$f(x)\; =\; \backslash frac,$$and similarly$$g(x)\; =\; \backslash frac,$$then$$f\backslash big(g(x)\backslash big)\; =\; g\backslash big(f(x)\backslash big)=\; \backslash frac.$$

More tersely stated, if for all

\alpha we let f _{\alpha} be what we called f above, then$$f\_\backslash alpha\; \backslash circ\; f\_\backslash beta\; =\; f\_.$$

If

x is the slope of a line, then f(x) is the slope of its rotation through an angle of - \alpha.

Relation to the complex exponential function

See main article: Euler's formula.

Euler's formula states that, for any real number x:^[30] $$e^\; =\; \backslash cos\; x\; +\; i\backslash sin\; x,$$where i is the imaginary unit. Substituting −x for x gives us:$$e^\; =\; \backslash cos(-x)\; +\; i\backslash sin(-x)\; =\; \backslash cos\; x\; -\; i\backslash sin\; x.$$

These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,^{[31] [32]} $$\backslash cos\; x\; =\; \backslash frac$$$$\backslash sin\; x\; =\; \backslash frac$$

These formulae are useful for proving many other trigonometric identities. For example, that means thatThat 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^[33]
\sin \theta = (e ^{i\theta} - e ^{-i\theta})/(2 i)	\arcsin x = - i ln \left(ix + \sqrt{ 1 - x ^{2}\right)}
\cos \theta = (e ^{i\theta} + e ^{-i\theta})/(2)	\arccos x = - i ln \left(x + \sqrt{x ^{2-1}\right)}
\tan \theta = - i (e ^{i\theta} - e ^{-i\theta})/(e ^{i\theta} + e ^{-i\theta})	\arctan x = (i)/(2) ln \left( (i + x)/(i - x) \right)
\csc \theta = (2 i)/(e ^{i\theta} - e ^{-i\theta})	\arccsc x = - i ln \left( (i)/(x) + \sqrt{ 1 - (1)/(x ^{2}) }\right)
\sec \theta = (2)/(e ^{i\theta} + e ^{-i\theta})	\arcsec x = - i ln \left( (1)/(x) + i \sqrt{ 1 - (1)/(x ^{2}) }\right)
\cot \theta = i (e ^{i\theta} + e ^{-i\theta})/(e ^{i\theta} - e ^{-i\theta})	\arccot x = (i)/(2) ln \left( (x - i)/(x + i) \right)
\operatorname{cis} \theta = e ^{i\theta}	\operatorname{arccis} x = - i ln x

Series expansion

When using a power series expansion to define trigonometric functions, the following identities are obtained:^[34]

$$\backslash sin\; x\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; \backslash cdots\; =\; \backslash sum\_^\; \backslash frac,$$$$\backslash cos\; x\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; =\; \backslash sum\_^\; \backslash frac.$$

Infinite product formulae

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:^{[35] [36]}

Inverse trigonometric functions

See main article: Inverse trigonometric functions.

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.

Taking the multiplicative inverse of both sides of the each equation above results in the equations for

\csc = (1)/(\sin) , \sec = (1)/(\cos) , and \cot = (1)/(\tan) . The right hand side of the formula above will always be flipped.For example, the equation for \cot(\arcsin x) is:$$\backslash cot(\backslash arcsin\; x)\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac$$while the equations for \csc(\arccos x) and \sec(\arccos x) are:$$\backslash csc(\backslash arccos\; x)\; =\; \backslash frac\; =\; \backslash frac\; \backslash qquad\; \backslash text\backslash quad\; \backslash sec(\backslash arccos\; x)\; =\; \backslash frac\; =\; \backslash frac.$$

The following identities are implied by the reflection identities. They hold whenever

x, r, s, - x, - r, and - s are in the domains of the relevant functions.

Also,^[37] $$\backslash arccos\; \backslash frac\; =\; \backslash arcsec\; x\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash arcsec\; \backslash frac\; =\; \backslash arccos\; x$$$$\backslash arcsin\; \backslash frac\; =\; \backslash arccsc\; x\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash arccsc\; \backslash frac\; =\; \backslash arcsin\; x$$

The arctangent function can be expanded as a series:$$\backslash arctan(nx)\; =\; \backslash sum\_^n\; \backslash arctan\backslash frac$$

Identities without variables

In terms of the arctangent function we have$$\backslash arctan\; \backslash frac\; =\; \backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac.$$

The curious identity known as Morrie's law,$$\backslash cos\; 20^\backslash circ\backslash cdot\backslash cos\; 40^\backslash circ\backslash cdot\backslash cos\; 80^\backslash circ\; =\; \backslash frac,$$

is a special case of an identity that contains one variable:$$\backslash prod\_^\backslash cos\backslash left(2^j\; x\backslash right)\; =\; \backslash frac.$$

Similarly,$$\backslash sin\; 20^\backslash circ\backslash cdot\backslash sin\; 40^\backslash circ\backslash cdot\backslash sin\; 80^\backslash circ\; =\; \backslash frac$$is a special case of an identity with

x = 20 ^{\circ} :$$\backslash sin\; x\; \backslash cdot\; \backslash sin\; \backslash left(60^\backslash circ\; -\; x\backslash right)\; \backslash cdot\; \backslash sin\; \backslash left(60^\backslash circ\; +\; x\backslash right)\; =\; \backslash frac.$$

For the case

x = 15 ^{\circ} ,

For the case

x = 10 ^{\circ} ,$$\backslash sin\; 10^\backslash circ\backslash cdot\backslash sin\; 50^\backslash circ\backslash cdot\backslash sin\; 70^\backslash circ\; =\; \backslash frac.$$

The same cosine identity is$$\backslash cos\; x\; \backslash cdot\; \backslash cos\; \backslash left(60^\backslash circ\; -\; x\backslash right)\; \backslash cdot\; \backslash cos\; \backslash left(60^\backslash circ\; +\; x\backslash right)\; =\; \backslash frac.$$

Similarly,

Similarly,

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):$$\backslash cos\; 24^\backslash circ\; +\; \backslash cos\; 48^\backslash circ\; +\; \backslash cos\; 96^\backslash circ\; +\; \backslash cos\; 168^\backslash circ\; =\; \backslash frac.$$

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:$$\backslash cos\; \backslash frac\; +\backslash cos\backslash left(2\backslash cdot\backslash frac\backslash right)\; +\backslash cos\backslash left(4\backslash cdot\backslash frac\backslash right)\; +\backslash cos\backslash left(5\backslash cdot\backslash frac\backslash right)\; +\backslash cos\backslash left(8\backslash cdot\backslash frac\backslash right)\; +\backslash cos\backslash left(10\backslash cdot\backslash frac\backslash right)=\; \backslash frac.$$

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 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:^[38] and so forth for all odd numbers, and hence$$\backslash cos\; \backslash frac+\backslash cos\; \backslash frac\; \backslash times\; \backslash cos\; \backslash frac\; +\; \backslash cos\; \backslash frac\; \backslash times\; \backslash cos\; \backslash frac\; \backslash times\; \backslash cos\; \backslash frac\; +\; \backslash dots\; =\; 1.$$

Many of those curious identities stem from more general facts like the following:$$\backslash prod\_^\; \backslash sin\backslash frac\; =\; \backslash frac$$and$$\backslash prod\_^\; \backslash cos\backslash frac\; =\; \backslash frac.$$

Combining these gives us$$\backslash prod\_^\; \backslash tan\backslash frac\; =\; \backslash frac$$

If is an odd number (

n = 2 m + 1 ) we can make use of the symmetries to get$$\backslash prod\_^\; \backslash tan\backslash frac\; =\; \backslash sqrt$$

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:$$\backslash prod\_^n\; \backslash sin\backslash frac\; =\; \backslash prod\_^\; \backslash cos\backslash frac\; =\; \backslash frac$$

Computing

An efficient way to compute 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:$$\backslash frac\; =\; 4\; \backslash arctan\backslash frac\; -\; \backslash arctan\backslash frac$$or, alternatively, by using an identity of Leonhard Euler:$$\backslash frac\; =\; 5\; \backslash arctan\backslash frac\; +\; 2\; \backslash arctan\backslash frac$$or by using Pythagorean triples:$$\backslash pi\; =\; \backslash arccos\backslash frac\; +\; \backslash arccos\backslash frac\; +\; \backslash arccos\backslash frac\; =\; \backslash arcsin\backslash frac\; +\; \backslash arcsin\backslash frac\; +\; \backslash arcsin\backslash frac.$$

Others include:^[39] $$\backslash frac\; =\; \backslash arctan\backslash frac\; +\; \backslash arctan\backslash frac,$$$$\backslash pi\; =\; \backslash arctan\; 1\; +\; \backslash arctan\; 2\; +\; \backslash arctan\; 3,$$$$\backslash frac\; =\; 2\backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac.$$

Generally, for numbers for which, let . This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are and its value will be in . In particular, the computed will be rational whenever all the values are rational. With these values,

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 values is not within . Note that if is rational, then the values in the above formulae are proportional to the Pythagorean triple .

For example, for terms,$$\backslash frac\; =\; \backslash arctan\backslash left(\backslash frac\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\backslash right)$$for any .

An identity of Euclid

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:$$\backslash sin^2\; 18^\backslash circ\; +\; \backslash sin^2\; 30^\backslash circ\; =\; \backslash sin^2\; 36^\backslash circ.$$

Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.

Composition of trigonometric functions

These identities involve a trigonometric function of a trigonometric function:^[40]

\cos(t \sin x) = J _{0(t)} + 2 \sum ^{inf ty} _{k =1} J _{2k} (t) \cos( 2 kx) \sin(t \sin x) = 2 \sum ^{inf ty} _{k =0} J _{2k+1} (t) \sin ( ( 2 k +1 )x ) \cos(t \cos x) = J _{0(t)} + 2 \sum ^{inf ty} _{k =1} ( -1 ) ^{kJ} _{2 k} (t) \cos( 2 kx) \sin(t \cos x) = 2 \sum ^{inf ty( -1 )} _{k =0} ^{k} J _{2k+1} (t) \cos ( ( 2 k +1 )x )

where are Bessel functions.

Further "conditional" identities for the case α + β + γ = 180°

A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.^[41] The following formulae apply to arbitrary plane triangles and follow from

\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).

Historical shorthands

See main article: 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

Dirichlet kernel

See main article: Dirichlet kernel. The Dirichlet kernel is the function occurring on both sides of the next identity:$$1\; +\; 2\backslash cos\; x\; +\; 2\backslash cos(2x)\; +\; 2\backslash cos(3x)\; +\; \backslash cdots\; +\; 2\backslash cos(nx)\; =\; \backslash frac.$$

The convolution of any integrable function of period

2 \pi with the Dirichlet kernel coincides with the function's n th-degree Fourier approximation. The same holds for any measure or generalized function.

Tangent half-angle substitution

See main article: Tangent half-angle substitution.

If we set $$t\; =\; \backslash tan\backslash frac\; x\; 2,$$ then^[42] $$\backslash sin\; x\; =\; \backslash frac;\backslash qquad\; \backslash cos\; x\; =\; \backslash frac;\backslash qquad\; e^\; =\; \backslash frac;\; \backslash qquad\; dx\; =\; \backslash frac,$$where

e ^{i} = \cos x + i \sin x, sometimes abbreviated to .

When this substitution of

t for is used in calculus, it follows that \sin x is replaced by, \cos x is replaced by and the differential is replaced by . Thereby one converts rational functions of \sin x and \cos x to rational functions of t in order to find their antiderivatives.

Viète's infinite product

See also: Viète's formula and Sinc function. $$\backslash cos\backslash frac\; \backslash cdot\; \backslash cos\; \backslash frac\backslash cdot\; \backslash cos\; \backslash frac\; \backslash cdots\; =\; \backslash prod\_^\backslash infty\; \backslash cos\; \backslash frac=\; \backslash frac\; =\; \backslash operatorname\; \backslash theta.$$

See also

• 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

Bibliography

• Book: Abramowitz. Milton. Milton Abramowitz. Stegun. Irene A.. Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications. New York. 978-0-486-61272-0. 1972.

External links

• Values of sin and cos, expressed in surds, for integer multiples of 3° and of °, and for the same angles csc and sec and tan
• Complete List of Trigonometric Formulas

Notes and References

1. Abramowitz and Stegun, p. 72, 4.3.13–15
2. Abramowitz and Stegun, p. 72, 4.3.7–9
3. Abramowitz and Stegun, p. 72, 4.3.16
4. Abramowitz and Stegun, p. 72, 4.3.17
5. Abramowitz and Stegun, p. 72, 4.3.18
6. Web site: Angle Sum and Difference Identities. www.milefoot.com. 2019-10-12.
7. Abramowitz and Stegun, p. 72, 4.3.19
8. Abramowitz and Stegun, p. 80, 4.4.32
9. Abramowitz and Stegun, p. 80, 4.4.33
10. Abramowitz and Stegun, p. 80, 4.4.34
11. Bronstein . Manuel . Simplification of real elementary functions . 207–211 . 10.1145/74540.74566 . Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation . G. H. . Gonnet . ISSAC '89 (Portland US-OR, 1989-07) . New York . . 1989 . 0-89791-325-6.
12. Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." The American Mathematical Monthly, volume 123, number 7, 701–703. https://doi.org/10.4169/amer.math.monthly.123.7.701
13. Hardy . Michael . 2016 . On Tangents and Secants of Infinite Sums . American Mathematical Monthly . 123 . 7 . 701–703 . 10.4169/amer.math.monthly.123.7.701 .
14. Web site: Sine, Cosine, and Ptolemy's Theorem .
15. Abramowitz and Stegun, p. 74, 4.3.48
16. Web site: Weisstein. Eric W.. Multiple-Angle Formulas. 2022-02-06. mathworld.wolfram.com. en.
17. Web site: Ward. Ken. Ken Ward's Mathematics Pages. Multiple angles recursive formula.
18. Abramowitz and Stegun, p. 72, 4.3.24–26
19. Abramowitz and Stegun, p. 72, 4.3.27–28
20. Abramowitz and Stegun, p. 72, 4.3.31–33
21. Book: Eves, Howard . An introduction to the history of mathematics . 1990 . Saunders College Pub . 0-03-029558-0 . 6th . Philadelphia . 309 . 20842510.
22. Abramowitz and Stegun, p. 72, 4.3.34–39
23. Warren P. . Johnson . Trigonometric Identities à la Hermite . . 117 . 4 . Apr 2010 . 311–327 . 10.4169/000298910x480784. 29690311 .
24. Web site: Product Identity Multiple Angle .
25. Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
26. Eddie . Ortiz Muñiz . Feb 1953 . 21 . 2 . A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities . American Journal of Physics . 140 . 10.1119/1.1933371 . 1953AmJPh..21..140M .
27. Book: Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems . illustrated . Ravi P. . Agarwal . Donal . O'Regan . Springer Science & Business Media . 2008 . 978-0-387-79146-3 . 185 . Extract of page 185
28. Book: Handbook of Mathematical Formulas and Integrals . 4th . Alan . Jeffrey . Hui-hui . Dai . Section 2.4.1.6 . 978-0-12-374288-9 . 2008 . Academic Press.
29. Fay . Temple H. . Kloppers . P. Hendrik . 2001 . The Gibbs' phenomenon . International Journal of Mathematical Education in Science and Technology . 32 . 1 . 73–89 . 10.1080/00207390117151.
30. Abramowitz and Stegun, p. 74, 4.3.47
31. Abramowitz and Stegun, p. 71, 4.3.2
32. Abramowitz and Stegun, p. 71, 4.3.1
33. Abramowitz and Stegun, p. 80, 4.4.26–31
34. Abramowitz and Stegun, p. 74, 4.3.65–66
35. Abramowitz and Stegun, p. 75, 4.3.89–90
36. Abramowitz and Stegun, p. 85, 4.5.68–69
37. Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
38. Humble . Steve . Grandma's identity . Mathematical Gazette . 88 . Nov 2004 . 524–525 . 10.1017/s0025557200176223. 125105552 .
39. Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39.
40. [Abramowitz and Stegun|Milton Abramowitz and Irene Stegun, ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'']
41. Er. K. C. Joshi, Krishna's IIT MATHEMATIKA. Krishna Prakashan Media. Meerut, India. page 636.
42. Abramowitz and Stegun, p. 72, 4.3.23