Trigonometric identity

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:

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

where $\sin^2 \theta$ means $^2$ and $\cos^2 \theta$ means $^2.$

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:

$\begin \sin\theta &= \pm \sqrt, \\ \cos\theta &= \pm \sqrt. \end$

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:
$\begin &1 + \cot^2\theta = \csc^2\theta \\ &1 + \tan^2\theta = \sec^2\theta \\ &\sec^2\theta + \csc^2\theta = \sec^2\theta\csc^2\theta \end$

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 $\theta,$ 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 an 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 $\theta^$ of this reflected line (vector) has the value
$\theta^ = 2 \alpha - \theta.$

The values of the trigonometric functions of these angles $\theta,\;\theta^$ 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 .

Shifts and periodicity

Signs

The sign of trigonometric functions depends on quadrant of the angle. If $< \theta \leq \pi$ and is the sign function,

\begin
\sgn(\sin \theta) = \sgn(\csc \theta) &= \begin
+1 & \text\ \ 0 < \theta < \pi \\
-1 & \text\ \ < \theta < 0 \\
0 & \text\ \ \theta \in \
\end
\\ mu\sgn(\cos \theta) = \sgn(\sec \theta) &= \begin
+1 & \text\ \ < \theta < \tfrac12\pi \\
-1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ \tfrac12\pi < \theta < \pi\\
0 & \text\ \ \theta \in \bigl\
\end
\\ mu\sgn(\tan \theta) = \sgn(\cot \theta) &= \begin
+1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ 0 < \theta < \tfrac12\pi \\
-1 & \text\ \ < \theta < 0 \ \ \text\ \ \tfrac12\pi < \theta < \pi \\
0 & \text\ \ \theta \in \bigl\
\end
\end

The trigonometric functions are periodic with common period $2\pi,$ so for values of outside the interval $(, \pi],$ they take repeating values (see above).

Angle sum and difference identities

These are also known as the (or ).
$\begin \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \\ \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta \end$

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.

Sines and cosines of sums of infinitely many angles

When the series $\sum_^\infty \theta_i$ absolute convergence, converges absolutely then

$\begin \biggl(\sum_^\infty \theta_i\biggl) &= \sum_ (-1)^\frac \!\! \sum_ \biggl(\prod_ \sin\theta_i \prod_ \cos\theta_i\biggr) \\ \biggl(\sum_^\infty \theta_i\biggr) &= \sum_ (-1)^\frac \, \sum_ \biggl(\prod_ \sin\theta_i \prod_ \cos\theta_i\biggr) . \end$

Because the series $\sum_^\infty \theta_i$ converges absolutely, it is necessarily the case that $\lim_ \theta_i = 0,$ $\lim_ \sin \theta_i = 0,$ and $\lim_ \cos \theta_i = 1.$ 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
$x_i = \tan \theta_i$
for $i = 0, 1, 2, 3, \ldots,$ that is,

\begin
e_0 &= 1 \\ pte_1 &= \sum_i x_i &&= \sum_i \tan\theta_i \\ pte_2 &= \sum_ x_i x_j &&= \sum_ \tan\theta_i \tan\theta_j \\ pte_3 &= \sum_ x_i x_j x_k &&= \sum_ \tan\theta_i \tan\theta_j \tan\theta_k \\
&\ \ \vdots &&\ \ \vdots
\end

Then

\begin
\Bigl(\sum_i \theta_i\Bigr)
&= \frac
\\ 0pt&= \frac

= \frac \\ 0pt
\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:
\begin
\tan(\theta_1 + \theta_2) &
= \frac
= \frac
= \frac,
\\ pt\tan(\theta_1 + \theta_2 + \theta_3) &
= \frac
= \frac,
\\ pt\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &
= \frac \\ pt&
= \frac,
\end

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

Secants and cosecants of sums

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

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. The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

\begin
\sec(\alpha+\beta+\gamma)
&= \frac
\\ pt\csc(\alpha+\beta+\gamma)
&= \frac
.
\end

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. 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 $\overline$ of length 1. Thus, the side $\overline = \sin \alpha$, $\overline = \cos \alpha$, $\overline = \sin \beta$ and $\overline = \cos \beta$.

By the inscribed angle theorem, the central angle subtended by the chord $\overline$ 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 $\frac$, so the length of $\overline$ is $2 \times \frac \sin(\alpha + \beta)$, 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 $, \overline, \cdot , \overline, =, \overline, \cdot , \overline, +, \overline, \cdot , \overline,$, 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 $\overline$ serve as a diameter instead of $\overline$.

Multiple-angle and half-angle formulae

Multiple-angle formulae

Double-angle formulae

Formulae for twice an angle.

* $\sin (2\theta) = 2 \sin \theta \cos \theta = (\sin \theta +\cos \theta)^2 - 1 = \frac$
* $\cos (2\theta) = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \frac$
* $\tan (2\theta) = \frac$
* $\cot (2\theta) = \frac = \frac$
* $\sec (2\theta) = \frac = \frac$
* $\csc (2\theta) = \frac = \frac$

Triple-angle formulae

Formulae for triple angles.

* $\sin (3\theta) =3\sin\theta - 4\sin^3\theta = 4\sin\theta\sin\left(\frac -\theta\right)\sin\left(\frac + \theta\right)$
* $\cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta =4\cos\theta\cos\left(\frac -\theta\right)\cos\left(\frac + \theta\right)$
* $\tan (3\theta) = \frac = \tan \theta\tan\left(\frac - \theta\right)\tan\left(\frac + \theta\right)$
* $\cot (3\theta) = \frac$
* $\sec (3\theta) = \frac$
* $\csc (3\theta) = \frac$

Multiple-angle formulae

Formulae for multiple angles.

* $\begin \sin(n\theta) &= \sum_ (-1)^\frac \cos^ \theta \sin^k \theta = \sin\theta\sum_^\sum_^ (-1)^ \cos^ \theta \\ &=\sin(\theta)\cdot\sum_^(-1)^k\cdot ^\cdot \\ &=2^ \prod_^ \sin(k\pi/n+\theta) \end$
* $\begin\cos(n\theta) &= \sum_ (-1)^\frac \cos^ \theta \sin^k \theta = \sum_^\sum_^ (-1)^ \cos^ \theta \\ &= \sum_^ (-1)^k\cdot ^\cdot \cdot\frac \end$
* $\cos((2n+1)\theta)=(-1)^n 2^\prod_^\cos(k\pi/(2n+1)-\theta)$
* $\cos(2 n \theta)=(-1)^n 2^ \prod_^ \cos((1+2k)\pi/(4n)-\theta)$
* $\tan(n\theta) = \frac$

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.

$\cos(nx)$ can be computed from $\cos((n-1)x)$, $\cos((n-2)x)$, and $\cos(x)$ with

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

This can be proved by adding together the formulae

$\begin \cos ((n-1)x + x) &= \cos ((n-1)x) \cos x-\sin ((n-1)x) \sin x \\ \cos ((n-1)x - x) &= \cos ((n-1)x) \cos x+\sin ((n-1)x) \sin x \end$

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
$\sin(nx)=2 \cos x \sin((n-1)x)-\sin((n-2)x)$
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:

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

Half-angle formulae

\begin
\sin \frac &= \sgn\left(\sin\frac\theta2\right) \sqrt \\ pt
\cos \frac &= \sgn\left(\cos\frac\theta2\right) \sqrt \\ pt
\tan \frac
&= \frac
= \frac
= \csc \theta - \cot \theta
= \frac \\ mu
&= \sgn(\sin \theta) \sqrt\frac
= \frac \\ pt
\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
\begin
\tan\frac &= \frac \\ pt \tan\left(\frac + \frac\right) &= \sec\theta + \tan\theta \\ pt \sqrt &= \frac
\end

Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

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.

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.

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.

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

The product-to-sum identities 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. 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

* \begin
\cos \theta\, \cos \varphi &= \tfrac12\bigl(\!\!~\cos(\theta - \varphi) + \cos(\theta + \varphi)\bigr) \\ mu\sin \theta\, \sin \varphi &= \tfrac12\bigl(\!\!~\cos(\theta - \varphi) - \cos(\theta + \varphi)\bigr) \\ mu\sin \theta\, \cos \varphi &= \tfrac12\bigl(\!\!~\sin(\theta + \varphi) + \sin(\theta - \varphi)\bigr) \\ mu\cos \theta\, \sin \varphi &= \tfrac12\bigl(\!\!~\sin(\theta + \varphi) - \sin(\theta - \varphi)\bigr)
\end
* $\tan \theta\, \tan \varphi =\frac$
* $\tan \theta\, \cot \varphi = \frac$
* \begin \prod_^n \cos \theta_k & = \frac\sum_ \cos(e_1\theta_1+\cdots+e_n\theta_n) \\ pt & \texte = (e_1,\ldots,e_n) \in S=\^n
\end
* $\prod_^n \sin\theta_k=\frac\begin \displaystyle\sum_\cos(e_1\theta_1+\cdots+e_n\theta_n)\prod_^n e_j \;\text\; n\; \text,\\ \displaystyle\sum_\sin(e_1\theta_1+\cdots+e_n\theta_n)\prod_^n e_j \;\text\; n\; \text \end$

Sum-to-product identities

The sum-to-product identities are as follows:

* $\sin \theta \pm \sin \varphi = 2 \sin\left( \frac \right) \cos\left( \frac \right)$
* $\cos \theta + \cos \varphi = 2 \cos\left( \frac \right) \cos\left( \frac \right)$
* $\cos \theta - \cos \varphi = -2\sin\left( \frac\right) \sin\left(\frac\right)$
* $\tan\theta\pm\tan\varphi=\frac$

Hermite's cotangent identity

Charles Hermite demonstrated the following identity. Suppose $a_1, \ldots, a_n$ are complex numbers, no two of which differ by an integer multiple of . Let

$A_ = \prod_ \cot(a_k - a_j)$

(in particular, $A_,$ being an empty product, is 1). Then

$\cot(z - a_1)\cdots\cot(z - a_n) = \cos\frac + \sum_^n A_ \cot(z - a_k).$

The simplest non-trivial example is the case :

$\cot(z - a_1)\cot(z - a_2) = -1 + \cot(a_1 - a_2)\cot(z - a_1) + \cot(a_2 - a_1)\cot(z - a_2).$

Finite products of trigonometric functions

For coprime integers ,

$\prod_^n \left(2a + 2\cos\left(\frac + x\right)\right) = 2\left( T_n(a)+^\cos(n x) \right)$

where is the Chebyshev polynomial.

The following relationship holds for the sine function

$\prod_^ \sin\left(\frac\right) = \frac.$

More generally for an integer

$\sin(nx) = 2^\prod_^ \sin\left(\frac\pi + x\right) = 2^\prod_^ \sin\left(\frac\pi - x\right).$

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

$\operatorname(nx) = \prod_^ \operatorname\left(\frac2\pi - x\right).$

This comes from the factorization of the polynomial $z^n - 1$ into linear factors (cf. root of unity): For any complex and an integer ,

$z^n - 1 = \prod_^\left( z - \exp\Bigl(\frac2\pi i\Bigr)\right).$

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,

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

where $c$ and $\varphi$ are defined as so:

$\begin c &= \sgn(a) \sqrt, \\ \varphi &= \bigl(\bigr), \end$

given that $a \neq 0.$

Arbitrary phase shift

More generally, for arbitrary phase shifts, we have

$a \sin(x + \theta_a) + b \sin(x + \theta_b)= c \sin(x+\varphi)$

where $c$ and $\varphi$ satisfy:

$\begin c^2 &= a^2 + b^2 + 2ab\cos \left(\theta_a - \theta_b \right) , \\ \tan \varphi &= \frac. \end$

More than two sinusoids

The general case reads
$\sum_i a_i \sin(x + \theta_i) = a \sin(x + \theta),$
where
$a^2 = \sum_a_i a_j \cos(\theta_i - \theta_j)$
and
$\tan\theta = \frac.$

Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases