Trigonometric functions

The following table summarizes the simplest algebraic values of trigonometric functions.[8] The symbol ∞ represents the point at infinity on the projectively extended real line; it is not signed, because, when it appears in the table, the corresponding trigonometric function tends to +∞ on one side, and to –∞ on the other side, when the argument tends to the value in the table.

${\displaystyle {\begin{array}{|c|ccccccccc|}\hline {\begin{matrix}{\text{Radian}}\\{\text{Degree}}\end{matrix}}&{\begin{matrix}0\\0^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{12}}\\15^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{8}}\\22.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{6}}\\30^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{4}}\\45^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{3}}\\60^{\circ }\end{matrix}}&{\begin{matrix}{\frac {3\pi }{8}}\\67.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {5\pi }{12}}\\75^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{2}}\\90^{\circ }\end{matrix}}\\\hline \sin &0&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&1\\\cos &1&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&0\\\tan &0&2-{\sqrt {3}}&{\sqrt {2}}-1&{\frac {\sqrt {3}}{3}}&1&{\sqrt {3}}&{\sqrt {2}}+1&2+{\sqrt {3}}&\infty \\\cot &\infty &2+{\sqrt {3}}&{\sqrt {2}}+1&{\sqrt {3}}&1&{\frac {\sqrt {3}}{3}}&{\sqrt {2}}-1&2-{\sqrt {3}}&0\\\sec &1&{\sqrt {6}}-{\sqrt {2}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}&2&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&{\sqrt {6}}+{\sqrt {2}}&\infty \\\csc &\infty &{\sqrt {6}}+{\sqrt {2}}&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&2&{\sqrt {2}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\sqrt {6}}-{\sqrt {2}}&1\\\hline \end{array}}}$

Sine and cosine are the unique differentiable functions such that

${\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin x&=\cos x,\\{\frac {d}{dx}}\cos x&=-\sin x,\\\sin 0&=0,\\\cos 0&=1.\end{aligned}}}$

Differentiating these equations, one gets that both sine and cosine are solutions of the differential equation

${\displaystyle y''+y=0.}$

Applying the quotient rule to the definition of the tangent as the quotient of the sine by the cosine, one gets that the tangent function verifies

${\displaystyle {\frac {d}{dx}}\tan x=1+\tan ^{2}x.}$

Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions[9]

${\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.\end{aligned}}}$

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form ${\textstyle (2k+1){\frac {\pi }{2}}}$ for the tangent and the secant, or ${\displaystyle k\pi }$ for the cotangent and the cosecant, where k is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.[10]

More precisely, defining

U_n, the nth up/down number,

B_n, the nth Bernoulli number, and

E_n, is the nth Euler number,

one has the following series expansions:[11]

${\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}\\&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}$

${\displaystyle {\begin{aligned}\csc x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}$

${\displaystyle {\begin{aligned}\sec x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}\\&{}=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}$

${\displaystyle {\begin{aligned}\cot x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}$

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:[12]

${\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}$

This identity can be proven with the Herglotz trick.[13] Combining the (–n)th with the nth term lead to absolutely convergent series:

${\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}\ ,\quad {\frac {\pi }{\sin \pi x}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}}.}$

The following infinite product for the sine is of great importance in complex anaylsis:

${\displaystyle \sin z=z\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}$

For the proof of this expansion, see Sine. From this, it can be deduced that

${\displaystyle \cos z=\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{\left(n-{\frac {1}{2}}\right)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}$

${\displaystyle \cos(\theta )}$ and ${\displaystyle \sin(\theta )}$ are the real and imaginary part of ${\displaystyle e^{i\theta }}$ respectively.

Euler's formula relates sine and cosine to the exponential function:

${\displaystyle e^{ix}=\cos x+i\sin x.}$

This formula is commonly considered for real values of x, but it remains true for all complex values.

Proof: Let ${\displaystyle f_{1}(x)=\cos x+i\sin x,}$ and ${\displaystyle f_{2}(x)=e^{ix}.}$ One has ${\textstyle {\frac {d}{dx}}f_{j}(x)=if_{j}(x)}$ for j = 1, 2. The quotient rule implies thus that ${\textstyle {\frac {d}{dx}}\left({\frac {f_{1}(x)}{f_{2}(x)}}\right)=0}$. Therefore, ${\textstyle {\frac {f_{1}(x)}{f_{2}(x)}}}$ is a constant function, which equals 1, as ${\displaystyle f_{1}(0)=f_{2}(0)=1.}$ This proves the formula.

One has

${\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}$

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:

${\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}$

When x is real, this may be rewritten as

${\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}$

Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity ${\displaystyle e^{a+b}=e^{a}e^{b}}$ for simplifying the result.

One can also define the trigonometric functions using various functional equations.

For example,[14] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula

${\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}$

and the added condition

${\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}$

The sine and cosine of a complex number ${\displaystyle z=x+iy}$ can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:

${\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}$

By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of ${\displaystyle z}$ becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

Trigonometric functions in the complex plane

${\displaystyle \sin z\,}$	${\displaystyle \cos z\,}$	${\displaystyle \tan z\,}$	${\displaystyle \cot z\,}$	${\displaystyle \sec z\,}$	${\displaystyle \csc z\,}$

The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:

${\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}$

All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has

${\displaystyle {\begin{aligned}\sin(x+2k\pi )&=\sin x\\\cos(x+2k\pi )&=\cos x\\\tan(x+k\pi )&=\tan x\\\cot(x+k\pi )&=\cot x\\\csc(x+2k\pi )&=\csc x\\\sec(x+2k\pi )&=\sec x.\end{aligned}}}$

The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is

${\displaystyle \sin ^{2}x+\cos ^{2}x=1.}$

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

Sum

${\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}$

Difference

${\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}$

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

${\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}$

These identities can be used to derive the product-to-sum identities.

By setting ${\displaystyle \theta =2x}$ and ${\displaystyle t=\tan x,}$ this allows expressing all trigonometric functions of ${\displaystyle \theta }$ as a rational fraction of ${\textstyle t=\tan {\frac {\theta }{2}}}$:

${\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}$

Together with

${\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}$

this is the tangent half-angle substitution, which allows reducing the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.

${\displaystyle {\begin{array}{|c|c|c|}\hline f(x)&f'(x)&\int f(x)\,dx\\\hline \sin x&\cos x&-\cos x+C\\\cos x&-\sin x&\sin x+C\\\tan x&\sec ^{2}x=1+\tan ^{2}x&-\ln \left(|\cos x|\right)+C\\\cot x&-\csc ^{2}x=-(1+\cot ^{2}x)&\ln \left(|\sin x|\right)+C\\\sec x&\sec x\tan x&\ln \left(|\sec x+\tan x|\right)+C\\\csc x&-\csc x\cot x&-\ln \left(|\csc x+\cot x|\right)+C\\\hline \end{array}}}$

In this sections A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

${\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}$

where Δ is the area of the triangle, or, equivalently,

${\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}$

where R is the triangle's circumradius.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,\,}$

or equivalently,

${\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}$

In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

The following all form the law of tangents[15]

${\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}\,;\qquad {\frac {\tan {\frac {A-C}{2}}}{\tan {\frac {A+C}{2}}}}={\frac {a-c}{a+c}}\,;\qquad {\frac {\tan {\frac {B-C}{2}}}{\tan {\frac {B+C}{2}}}}={\frac {b-c}{b+c}}.}$

The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences, for the lengths and corresponding opposite angles, are apparent in the theorem.

If

${\displaystyle \zeta ={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}~}$ (the radius of the inscribed circle for the triangle)

and

${\displaystyle s={\frac {a+b+c}{2}}~}$ (the semi-perimeter for the triangle),

then the following all form the law of cotangents[15]

${\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{\zeta }}~;\qquad \cot {\frac {B}{2}}={\frac {s-b}{\zeta }}~;\qquad \cot {\frac {C}{2}}={\frac {s-c}{\zeta }}~.}$

It follows that

${\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{\zeta }}~.}$

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.

A Lissajous curve, a figure formed with a trigonometry-based function.

An animation of the additive synthesis of a square wave with an increasing number of harmonics

Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.[16]

Under rather general conditions, a periodic function f(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series.[17] Denoting the sine or cosine basis functions by φ_k, the expansion of the periodic function f(t) takes the form:

${\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}$

For example, the square wave can be written as the Fourier series

${\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}$

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.