Trigonometric functions

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Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

The most familiar trigonometric functions are the sine, the cosine, and the tangent. Their reciprocal are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

Right-angled triangle definitions

Top: Trigonometric function sin θ for selected angles θ, πθ, π + θ, and 2πθ in the four quadrants.
Bottom: Graph of sine function versus angle. Angles from the top panel are identified.

Plot of the six trigonometric functions and the unit circle for an angle of 0.7 radians

In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lover case letter denotes an edge of the triangle and its length.

Given an acute angle A of a right-angled triangle (see figure) the hypotenuse h is the side that connects the two acute angles. The side b adjacent to A is the side of the triangle that connects A to the right angle. The third side a is said opposite to A.

If the angle A is given, then all sides of the right-angled triangle are well defined up to a scaling factor. This means that the ratio of any two side lengths depends only on A. These six ratios define thus six functions of A, which are the trigonometric functions. More precisely, the six trigonometric functions are:

• sine: ${\displaystyle \sin A={\frac {a}{h}}={\frac {\mathrm {opposite} }{\mathrm {hypothenuse} }}}$
• cosine: ${\displaystyle \cos A={\frac {b}{h}}={\frac {\mathrm {adjacent} }{\mathrm {hypothenuse} }}}$
• tangent: ${\displaystyle \tan A={\frac {a}{b}}={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$
• cosecant: ${\displaystyle \csc A={\frac {h}{a}}={\frac {\mathrm {hypothenuse} }{\mathrm {opposite} }}}$
• secant: ${\displaystyle \sec A={\frac {h}{b}}={\frac {\mathrm {hypothenuse} }{\mathrm {adjacent} }}}$
• cotangent: ${\displaystyle \cot A={\frac {b}{a}}={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}$

In a right angled triangle, the sum of the two acute angles is a right angle, that is 90° or ${\displaystyle {\frac {\pi }{2}}}$  radians. This induces relationships between trigonometric functions that are summarized in the following table, where the angle is denoted by ${\displaystyle \theta }$  instead of A.

Function Abbreviation Description Relationship (using radians)
sine sin opposite/hypotenuse ${\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}$
cosine cos adjacent/hypotenuse ${\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}$
tangent tan (or tg) opposite/adjacent ${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}$
cotangent cot (or cotan or cotg or ctg or ctn) adjacent/opposite ${\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}$
secant sec hypotenuse/adjacent ${\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}$
cosecant csc (or cosec) hypotenuse/opposite ${\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}$

In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees.

When using trigonometric function in calculus, their argument is generally not an angle, but rather a real number. In this case, it is more suitable to express the argument of the trigonometric as the length of the arc of the unit circle delimited by an angle with the center of the circle as vertex. Therefore, one uses the radian as angular unit: a radian is the angle that delimites an arc of length 1 on the unit circle. A complete turn is thus an angle of 2π radians.

A great advantage of radians is that many formulas are much simpler when using them, typically all formulas relative to derivatives and integrals.

This is thus a general convention that, when the angular unit is not explicitly specified, the arguments of trigonometric functions are always expressed in radians.

Unit-circle definitions

In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.

Signs of trigonometric functions in each quadrant. The mnemonic "all science teachers (are) crazy" lists the functions which are positive from quadrants I to IV.[3] This is a variation on the mnemonic "All Students Take Calculus".

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions permit the definition of the trigonometric functions for angles between 0 and ${\displaystyle {\frac {\pi }{2}}}$  radian (90°), the unit circle definitions allow to extend the domain of the trigonometric functions to all positive and negative real numbers.

Rotating a ray from the direction of the positive half of the x-axis by an angle θ (counterclockwise for ${\displaystyle \theta >0,}$  and clockwise for ${\displaystyle \theta <0}$ ) yields intersection points of this ray (see the figure) with the unit circle: ${\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} })}$ , and, by extending the ray to a line if necessary, with the line ${\displaystyle {\text{“}}x=1{\text{”}}:\;\mathrm {B} =(x_{\mathrm {B} },y_{\mathrm {B} }),}$  and with the line ${\displaystyle {\text{“}}y=1{\text{”}}:\;\mathrm {C} =(x_{\mathrm {C} },y_{\mathrm {C} }).}$  The tangent line to the unit circle in point A, which is orthogonal to this ray, intersects the y- and x-axis in points ${\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}$  and ${\displaystyle \mathrm {E} =(x_{\mathrm {E} },0)}$ . The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of θ in the following manner.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A, i.e.,

${\displaystyle \cos(\theta )=x_{\mathrm {A} }\quad }$  and ${\displaystyle \quad \sin(\theta )=y_{\mathrm {A} }.}$ [4]

In the range ${\displaystyle 0\leq \theta \leq \pi /2}$  this definition coincides with the right-angled triangle definition by taking the right-angled triangle to have the unit radius OA as hypotenuse, and since for all points ${\displaystyle \mathrm {P} =(x,y)}$  on the unit circle the equation ${\displaystyle x^{2}+y^{2}=1}$  holds, this definition of cosine and sine also satisfies the Pythagorean identity

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

The other trigonometric functions can be found along the unit circle as

${\displaystyle \tan(\theta )=y_{\mathrm {B} }\quad }$  and ${\displaystyle \quad \cot(\theta )=x_{\mathrm {C} },}$
${\displaystyle \csc(\theta )\ =y_{\mathrm {D} }\quad }$  and ${\displaystyle \quad \sec(\theta )=x_{\mathrm {E} }.}$

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}$

Trigonometric functions: Sine, Cosine, Tangent, Cosecant (dotted), Secant (dotted), Cotangent (dotted)

As a rotation of an angle of ${\displaystyle \pm 2\pi }$  does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of ${\displaystyle 2\pi }$ . Thus trigonometric functions are periodic functions with period ${\displaystyle 2\pi }$ . That is, the equalities

${\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }$  and ${\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}$

hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. Observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, shows that 2π is the smallest value for which they are periodic, i.e., 2π is the fundamental period of these functions. However, already after a rotation by an angle ${\displaystyle \pi }$  the points B and C return to their original position, so that the tangent function and the cotangent function have a fundamental period of π. That is, the equalities

${\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }$  and ${\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}$

hold for any angle θ and any integer k.

Algebraic values

The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

The algebraic expressions for sin 0, sin π/6 = sin 30°, sin π/4 = sin 45°, sin π/3 = sin 60° and sin π/2 = sin 90° are

${\displaystyle 0,\quad {\frac {1}{2}},\quad {\frac {\sqrt {2}}{2}},\quad {\frac {\sqrt {3}}{2}},\quad 1,}$

respectively. Writing the numerators as square roots of consecutive natural numbers ${\displaystyle {\frac {\sqrt {0}}{2}},{\frac {\sqrt {1}}{2}},{\frac {\sqrt {2}}{2}},{\frac {\sqrt {3}}{2}},{\frac {\sqrt {4}}{2}}}$  provides an easy way to remember the values.[5]

Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle. For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots, see Trigonometric constants expressed in real radicals. These values of the sine and the cosine may thus be constructed by ruler and compass.

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows prooving that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

For an angle which, measured in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.

In calculus

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Animation for the approximation of cosine via Taylor polynomials.

${\displaystyle \cos(x)}$  together with the first Taylor polynomials ${\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}$

Trigonometric function are differentiable. This is not immediately evident from the above geometrical definitions. Moreover, the modern trends, in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined with calculus method

For defining trigonometric functions inside calculus, there are two equivalent possibilities, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.

Definition by differential equations

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

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

Power series expansion

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[6]

{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\\end{aligned}}}
{\displaystyle {\begin{aligned}\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}\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 ${\displaystyle (2k+1){\frac {\pi }{2}}}$  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 meromorphic trigonometric function, which are harder to solve. Defining

Un, the nth up/down number,
Bn, the nth Bernoulli number, and
En, is the nth Euler number,

one gets the series expansions, which a finite radius of convergence: [7]

{\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}}}

When the series for the tangent and secant functions are expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "tangent numbers" and "secant numbers" respectively, have a combinatorial interpretation: they enumerate alternating permutations of finite sets, of odd cardinality for the tangent series and even cardinality for the secant series.[8] The series itself can be found by a power series solution of the aforementioned differential equation.

From a theorem in complex analysis, there is a unique analytic continuation of this real function to the domain of complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.

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:[9]

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

This identity can be proven with the Herglotz trick.[10] Combining the (–n)th with the nth term lead to absolutely convergent series:

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

Relationship to exponential function and complex numbers

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

It can be shown from the series definitions[11] that the sine and cosine functions are respectively the imaginary and real parts of the exponential function of a purely imaginary argument. That is, if x is real, we have

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

and

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

The latter identity, although primarily established for real x, remains valid for every complex x, and is called Euler's formula.

Euler's formula can be used to derive most trigonometric identities from the properties of the exponential function, by writing sine and cosine as:

${\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},}$

and

${\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}\,.}$

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.

{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin x\cosh y+i\cos x\sinh y\,,\\\cos(x+iy)&=\cos x\cosh y-i\sin x\sinh y\,.\end{aligned}}}

This exhibits a deep relationship between the complex sine and cosine functions and their real (sin, cos) and hyperbolic real (sinh, cosh) counterparts.

Complex graphs

In the following graphs the domain is the complex plane pictured with domain coloring, and the range values are indicated at each point by color. Brightness indicates the size (absolute value) of the range value, with black being zero. Hue varies with argument, or angle, measured from the positive real axis.

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

Definitions via differential equations

Both the sine and cosine functions satisfy the linear differential equation:

${\displaystyle y''=-y\,.}$

That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional function space V consisting of all solutions of this equation,

• the sine function is the unique solution satisfying the initial condition ${\displaystyle \left(y'(0),y(0)\right)=(1,0)\,}$  and
• the cosine function is the unique solution satisfying the initial condition ${\displaystyle \left(y'(0),y(0)\right)=(0,1)\,}$ .

Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions.

Further, the observation that sine and cosine satisfies y″ = −y means that they are eigenfunctions of the second-derivative operator.

The tangent function is the unique solution of the nonlinear differential equation

${\displaystyle y'=1+y^{2}\,}$

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation.[12]

Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

${\displaystyle f(x)=\sin(kx),\,}$

then the derivatives will scale by amplitude.

${\displaystyle f'(x)=k\cos(kx).\,}$

Here, k is a constant that represents a mapping between units. If x is in degrees, then

${\displaystyle k={\frac {\pi }{180^{\circ }}}.}$

This means that the second derivative of a sine in degrees does not satisfy the differential equation

${\displaystyle y''=-y\,}$

but rather

${\displaystyle y''=-k^{2}y.\,}$

The cosine's second derivative behaves similarly.

This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

Basic identities

Many identities interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity is written

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

which is standard shorthand notation for

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

Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines 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}}}

These in turn lead to the following three-angle formulae:

{\displaystyle {\begin{aligned}\sin \left(x+y+z\right)&=\sin x\cos y\cos z+\sin y\cos z\cos x+\sin z\cos y\cos x-\sin x\sin y\sin z,\\\cos \left(x+y+z\right)&=\cos x\cos y\cos z-\cos x\sin y\sin z-\cos y\sin x\sin z-\cos z\sin x\sin y,\\\tan(x+y+z)&={\frac {\tan x+\tan y+\tan z-\tan x\tan y\tan z}{1-\tan x\tan y-\tan y\tan z-\tan z\tan x}}\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,\\\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x,\\\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}

When three angles are equal, the three-angle formulae simplify to

{\displaystyle {\begin{aligned}\sin 3x&=3\sin x-4\sin ^{3}x,\\\cos 3x&=4\cos ^{3}x-3\cos x,\\\tan 3x&={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}.\end{aligned}}}

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

Calculus

For integrals and derivatives of trigonometric functions, see the relevant sections of Differentiation of trigonometric functions, Lists of integrals and List of integrals of trigonometric functions. Below is the list of the derivatives and integrals of the six basic trigonometric functions. The number C is a constant of integration.

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

Definitions using functional equations

In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the difference formula. Taking as given these formulas, one can prove that only two continuous functions satisfy those conditions. Formally, there exists exactly one pair of continuous functions—sin and cos—such that for all real numbers x and y, the following equation holds:[13]

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

${\displaystyle 0

This may also be used for extending sine and cosine to the complex numbers. Other functional equations are also possible for defining trigonometric functions.

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. This section, however, describes details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found.

The first step in computing any trigonometric function is range reduction—reducing the given angle to a "reduced angle" inside a small range of angles, say 0 to π/2, using the periodicity and symmetries of the trigonometric functions.

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History below), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2) = 1).

Modern computers use a variety of techniques.[14] One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup—they first look up the closest angle in a small table, and then use the polynomial to compute the correction.[15] Devices that lack hardware multipliers often use an algorithm called CORDIC (as well as related techniques), which uses only addition, subtraction, bitshift, and table lookup. These methods are commonly implemented in hardware floating-point units for performance reasons.

For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral.[16]

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. For example, the sine, cosine and tangent of any integer multiple of π/60 radians (3°) can be found exactly by hand.

Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45°). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π/4 radians (45°) can then be found using the Pythagorean theorem:

${\displaystyle c={\sqrt {a^{2}+b^{2}}}={\sqrt {2}}\,.}$

Therefore:

${\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }=\cos {\frac {\pi }{4}}=\cos 45^{\circ }={1 \over {\sqrt {2}}}={{\sqrt {2}} \over 2},\,}$
${\displaystyle \tan {\frac {\pi }{4}}=\tan 45^{\circ }={{\sin {\frac {\pi }{4}}} \over {\cos {\frac {\pi }{4}}}}={1 \over {\sqrt {2}}}\cdot {{\sqrt {2}} \over 1}={{\sqrt {2}} \over {\sqrt {2}}}=1.\,}$

Computing trigonometric functions from an equilateral triangle

To determine the trigonometric functions for angles of π/3 radians (60°) and π/6 radians (30°), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60°). By dividing it into two, we obtain a right triangle with π/6 radians (30°) and π/3 radians (60°) angles. For this triangle, the shortest side is 1/2, the next largest side is 3/2 and the hypotenuse is 1. This yields:

${\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }=\cos {\frac {\pi }{3}}=\cos 60^{\circ }={1 \over 2}\,,}$
${\displaystyle \cos {\frac {\pi }{6}}=\cos 30^{\circ }=\sin {\frac {\pi }{3}}=\sin 60^{\circ }={{\sqrt {3}} \over 2}\,,}$
${\displaystyle \tan {\frac {\pi }{6}}=\tan 30^{\circ }=\cot {\frac {\pi }{3}}=\cot 60^{\circ }={1 \over {\sqrt {3}}}={{\sqrt {3}} \over 3}\,.}$

Special values in trigonometric functions

There are some commonly used special values in trigonometric functions, as shown in the following table.

${\displaystyle {\begin{array}{|c|cccccccc|}{\begin{matrix}{\text{Radian}}\\{\text{Degree}}\end{matrix}}&{\begin{matrix}0\\0^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{12}}\\15^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{8}}\\22.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{6}}\\30^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{4}}\\45^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{3}}\\60^{\circ }\end{matrix}}&{\begin{matrix}{\frac {5\pi }{12}}\\75^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{2}}\\90^{\circ }\end{matrix}}\\\hline \sin &0&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&1\\\cos &1&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {1}{2}}&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&0\\\tan &0&2-{\sqrt {3}}&{\sqrt {2}}-1&{\frac {\sqrt {3}}{3}}&1&{\sqrt {3}}&2+{\sqrt {3}}&\infty \\\cot &\infty &2+{\sqrt {3}}&{\sqrt {2}}+1&{\sqrt {3}}&1&{\frac {\sqrt {3}}{3}}&2-{\sqrt {3}}&0\\\sec &1&{\sqrt {6}}-{\sqrt {2}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}&2&{\sqrt {6}}+{\sqrt {2}}&\infty \\\csc &\infty &{\sqrt {6}}+{\sqrt {2}}&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&2&{\sqrt {2}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {6}}-{\sqrt {2}}&1\\\end{array}}}$ [17]

The symbol here represents the point at infinity on the projectively extended real line, the limit on the extended real line is +∞ on one side and -∞ on the other.

Inverse functions

The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function. Therefore, to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:

${\displaystyle {\begin{array}{rrc}{\text{Function}}&{\text{Definition}}&{\text{Value Field}}\\\hline \arcsin x=y&\sin y=x&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}\\\arccos x=y&\cos y=x&0\leq y\leq \pi \\\arctan x=y&\tan y=x&-{\frac {\pi }{2}}

The notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" for arcsecant can be confused with "arcsecond".

Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,

${\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \,.}$

These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:

${\displaystyle \arcsin z=\int _{0}^{z}{\frac {1}{\sqrt {1-x^{2}}}}\,dx,\quad |z|<1.}$

Analogous formulas for the other functions can be found at inverse trigonometric functions. Using the complex logarithm, one can generalize all these functions to complex arguments:

{\displaystyle {\begin{aligned}\arcsin z&=-i\log \left(iz+{\sqrt {1-z^{2}}}\right),\,\\\arccos z&=-i\log \left(z+{\sqrt {z^{2}-1}}\right),\,\\\arctan z&={\frac {1}{2}}i\log \left({\frac {1-iz}{1+iz}}\right).\end{aligned}}}

Connection to the inner product

In an inner product space, the angle between two non-zero vectors is defined to be

${\displaystyle \operatorname {angle} (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\cdot \|y\|}}.}$

Applications

Angles and sides of a triangle

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.

Law of sines

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.

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

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.

Law of cosines

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.

Law of tangents

The following all form the law of tangents[18]

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

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.

Law of cotangents

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[18]

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

It follows that

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

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.

Periodic functions

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.[19]

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.[20] Denoting the sine or cosine basis functions by φk, 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.

History

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

The functions sine and cosine can be traced to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[21]

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.[22] Al-Khwārizmī produced tables of sines, cosines and tangents. They were studied by authors including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series.[23]

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).[24]

The first published use of the abbreviations sin, cos, and tan is probably by the 16th century French mathematician Albert Girard.[citation needed]

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.[25]

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).[21]

A few functions were common historically, but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables[21]), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin2(θ/2)),[26] the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec(π/2θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.

Etymology

The word sine derives[27] from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.[28] The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".[29]

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.[30]

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.[31][32]

Notes

1. ^ Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.
2. ^ Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. ISBN 978-0-48643480-3. Archived from the original on 2018-02-15. Retrieved 2017-08-13.
3. ^ Heng, Cheng and Talbert, "Additional Mathematics" Archived 2015-03-20 at the Wayback Machine, page 228
4. ^ Bityutskov, V.I. (2011-02-07). "Trigonometric Functions". Encyclopedia of Mathematics. Archived from the original on 2017-12-29. Retrieved 2017-12-29.
5. ^ Larson, Ron (2013). Trigonometry (9th ed.). Cengage Learning. p. 153. ISBN 978-1-285-60718-4. Archived from the original on 2018-02-15. Extract of page 153 Archived 2018-02-15 at the Wayback Machine
6. ^ See Ahlfors, pages 43–44.
7. ^ Abramowitz; Weisstein.
8. ^ Stanley, Enumerative Combinatorics, Vol I., page 149
9. ^ Aigner, Martin; Ziegler, Günter M. (2000). Proofs from THE BOOK (Second ed.). Springer-Verlag. p. 149. ISBN 978-3-642-00855-9. Archived from the original on 2014-03-08.
10. ^ Remmert, Reinhold (1991). Theory of complex functions. Springer. p. 327. ISBN 978-0-387-97195-7. Archived from the original on 2015-03-20. Extract of page 327 Archived 2015-03-20 at the Wayback Machine
11. ^ For a demonstration, see Euler's formula#Using power series
12. ^ Needham, Tristan (1998). Visual Complex Analysis. ISBN 978-0-19-853446-4.
13. ^ Kannappan, Palaniappan (2009). Functional Equations and Inequalities with Applications. Springer. ISBN 978-0387894911.
14. ^ Kantabutra.
15. ^ However, doing that while maintaining precision is nontrivial, and methods like Gal's accurate tables, Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used.
16. ^ Brent, Richard P. (April 1976). "Fast Multiple-Precision Evaluation of Elementary Functions". J. ACM. 23 (2): 242–251. doi:10.1145/321941.321944. ISSN 0004-5411.
17. ^ Abramowitz, Milton and Irene A. Stegun, p.74
18. ^ a b The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 529-530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.
19. ^ Farlow, Stanley J. (1993). Partial differential equations for scientists and engineers (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. ISBN 978-0-486-67620-3. Archived from the original on 2015-03-20.
20. ^ See for example, Folland, Gerald B. (2009). "Convergence and completeness". Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77ff. ISBN 978-0-8218-4790-9. Archived from the original on 2015-03-19.
21. ^ a b c Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7, p. 210.
22. ^ Gingerich, Owen (1986). "Islamic Astronomy". 254. Scientific American: 74. Archived from the original on 2013-10-19. Retrieved 2010-07-13.
23. ^ O'Connor, J. J.; Robertson, E. F. "Madhava of Sangamagrama". School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 2006-05-14. Retrieved 2007-09-08.
24. ^ "Fincke biography". Archived from the original on 2017-01-07. Retrieved 2017-03-15.
25. ^ Bourbaki, Nicolás (1994). Elements of the History of Mathematics. Springer.
26. ^ Nielsen (1966, pp. xxiii–xxiv)
27. ^ The anglicized form is first recorded in 1593 in Thomas Fale's Horologiographia, the Art of Dialling.
28. ^ Various sources credit the first use of sinus to either
See Merlet, A Note on the History of the Trigonometric Functions in Ceccarelli (ed.), International Symposium on History of Machines and Mechanisms, Springer, 2004
See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
See Katx, Victor (July 2008). A history of mathematics (3rd ed.). Boston: Pearson. p. 210 (sidebar). ISBN 978-0321387004.
29. ^ See Plofker, Mathematics in India, Princeton University Press, 2009, p. 257
See "Clark University". Archived from the original on 2008-06-15.
See Maor (1998), chapter 3, regarding the etymology.
30. ^ Oxford English Dictionary
31. ^ Gunter, Edmund (1620). Canon triangulorum.
32. ^ Roegel, Denis, ed. (2010-12-06). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 2017-07-28. Retrieved 2017-07-28.