# Trigonometric functions

The class of elementary functions sine, cosine, tangent, cotangent, secant, cosecant. These are denoted, respectively, by: $\sin x$, $\cos x$, $\mathop{\rm tan} x$( or $\mathop{\rm tg} x$), $\mathop{\rm cot} x$( or $\mathop{\rm cotan} x$), $\mathop{\rm sec} x$, $\mathop{\rm csc} x$( or $\cosec x$).

## Trigonometric functions of a real argument.

Let $\alpha$ be a real number. Let $A = ( x _ \alpha , y _ \alpha )$ be the end point of the arc on the unit circle $x ^ {2} + y ^ {2} = 1$( see Fig. a) having initial point $B = ( 1, 0)$ and length $| \alpha |$. The arc from $B$ to $A$ is taken in the counter-clockwise direction if $\alpha \geq 0$, and in the clockwise direction if $\alpha < 0$. If $\alpha = 0$, then $A = B$; if, e.g., $\alpha = (- 7 \pi )/ ( 2)$, then $A = ( 0, 1)$. $B$, $| BA | = \alpha > 0$

Figure: t094210a

The basic trigonometric functions sine and cosine are defined at $\alpha$ by the formulas

$$\sin \alpha = \ y _ \alpha ,\ \ \cos \alpha = \ x _ \alpha .$$

The remaining trigonometric functions can be defined by the formulas

$$\mathop{\rm tan} \alpha = \ \frac{\sin \alpha }{\cos \alpha } ,\ \ \mathop{\rm cot} \alpha = \ \frac{\cos \alpha }{\sin \alpha } ,$$

$$\mathop{\rm sec} \alpha = { \frac{1}{\cos \alpha } } ,\ \ \mathop{\rm csc} \alpha = { \frac{1}{\sin \alpha } } .$$

All trigonometric functions are periodic. The graphs of the trigonometric functions are given in Fig. b.

Figure: t094210b

The main properties of the trigonometric functions — the domain of definition, the range, the parity, and sections of monotonicity — are given in the table below.

<tbody> </tbody>
 Function Domain of definition Range of values Parity Section of monotonicity $\sin x$ $- \infty < x <+ \infty$ $[- 1, + 1]$ Odd $\begin{array}{c} \textrm{ increases for } x \in (( 4n - 1) \pi /2, ( 4n + 1) \pi /2) \\ \textrm{ decreases for } x \in (( 4n + 1) \pi /2, ( 4n + 3) \pi /2) \end{array}$ $\cos x$ $- \infty < x <+\infty$ $[- 1, + 1]$ Even $\begin{array}{c} \textrm{ increases for } x \in (( 2n - 1) \pi , 2n \pi) \\ \textrm{ decreases for } x \in ( 2n \pi , ( 2n + 1) \pi ) \end{array}$ $\mathop{\rm tan} x$ $x \neq \pi n + \pi / 2$ $(- \infty , + \infty )$ Odd increases for $x \in (( 2n - 1) \pi /2, ( 2n + 1) \pi /2)$ $\mathop{\rm cot} x$ $x \neq \pi n$ $(- \infty , + \infty )$ Odd decreases for $x \in ( n \pi , ( n + 1) \pi )$ $\mathop{\rm sec} x$ $x \neq \pi n + \pi / 2$ $(- \infty , - 1 ] \cup [ + 1, + \infty )$ Even $\begin{array}{c} \textrm{ increases for } x \in ( 2n \pi , ( 2n + 1) \pi ) \\ \textrm{ decreases for } x \in (( 2n - 1) \pi , 2n \pi ) \end{array}$ $\mathop{\rm csc} x$ $x \neq \pi n$ $(- \infty , - 1 ] \cup [ + 1, + \infty )$ Odd $\begin{array}{c} \textrm{ increases for } x \in (( 4n + 1) \pi /2, ( 4n + 3) \pi /2) \\ \textrm{ decreases for } x \in (( 4n - 1) \pi /2, ( 4n + 1) \pi /2) \end{array}$

Each trigonometric function is continuous and infinitely differentiable at each point of its domain of definition; the derivatives of the trigonometric functions are:

$$( \sin x) ^ \prime = \cos x,\ \ ( \cos x) ^ \prime = - \sin x,$$

$$( \mathop{\rm tan} x) ^ \prime = { \frac{1}{\cos ^ {2} x } } ,\ ( \mathop{\rm cot} x) ^ \prime = - { \frac{1}{\sin ^ {2} x } } .$$

The integrals of the trigonometric functions are:

$$\int\limits \sin x dx = - \cos x + C,\ \ \int\limits \cos x dx = \sin x + C,$$

$$\int\limits \mathop{\rm tan} x dx = - \mathop{\rm ln} | \cos x | + C, \int\limits \mathop{\rm cot} x dx = \mathop{\rm ln} | \sin x | + C.$$

All trigonometric functions have a power series expansion:

$$\sin x = x - \frac{x ^ {3} }{3! } + \frac{x ^ {5} }{5! } - \dots + (- 1) ^ {n} \frac{x ^ {2n + 1 } }{( 2n + 1)! } + \dots$$

for $| x | < \infty$;

$$\cos x = 1 - \frac{x ^ {2} }{2! } + \frac{x ^ {4} }{4! } - \frac{x ^ {6} }{6! } + \dots + (- 1) ^ {n} \frac{x ^ {2n} }{( 2n)! } + \dots$$

for $| x | < \infty$;

$$\mathop{\rm tan} x = \ x + { \frac{1}{3} } x ^ {3} + { \frac{2}{15} } x ^ {5} + { \frac{17}{315} } x ^ {7} + \dots$$

$$\dots + \frac{2 ^ {2n} ( 2 ^ {2n} - 1) | B _ {n} | }{( 2n)! } x ^ {2n - 1 } + \dots$$

for $| x | < {\pi / 2 }$;

$$\mathop{\rm cot} x = { \frac{1}{x} } -$$

$$- \left [ { \frac{x}{3} } + \frac{x ^ {3} }{45 } + \frac{2x ^ {5} }{945 } + \frac{x ^ {7} }{4725 } + \dots + \frac{2 ^ {2n} | B _ {n} | }{( 2n)! } x ^ {2n - 1 } + \dots \right ]$$

for $0 < | x | < \pi$( the $B _ {n}$ are the Bernoulli numbers).

The function inverse to the function $x = \sin y$ defines $y$ as a many-valued function of $x$, it is denoted by $y = \mathop{\rm arc} \sin x$. The inverse functions of the other trigonometric functions are defined similarly; they are all called inverse trigonometric functions.

## Trigonometric functions of a complex variable.

The trigonometric functions for complex values of the variable $z = x + iy$ are defined as analytic continuations (cf. Analytic continuation) of the corresponding trigonometric functions of the real variable in the complex plane.

Thus, $\sin z$ and $\cos z$ can be defined by means of the power series for $\sin x$ and $\cos x$ given above. These series converge in the entire complex plane, therefore $\sin z$ and $\cos z$ are entire functions (cf. Entire function).

The trigonometric functions tangent and cotangent are defined by the formulas

$$\mathop{\rm tan} z = \ \frac{\sin z }{\cos z } ,\ \ \mathop{\rm cot} z = \ \frac{\cos z }{\sin z } .$$

The trigonometric functions $\mathop{\rm tan} z$ and $\mathop{\rm cot} z$ are meromorphic functions (cf. Meromorphic function). The poles of $\mathop{\rm tan} z$ are simple (of order one) and are situated at the points $z = \pi / 2 + \pi n$, $n = 0, \pm 1 ,\dots$.

All formulas for the trigonometric functions of a real argument remain true for a complex argument as well.

In contrast to the trigonometric functions of a real variable, the functions $\sin z$ and $\cos z$ take all complex values: The equations $\sin z = a$ and $\cos z = a$ each have infinitely many solutions for any complex $a$:

$$z = \mathop{\rm arc} \sin a = \ - i \mathop{\rm ln} ( ia \pm \sqrt {1 - a ^ {2} } ),$$

$$z = \mathop{\rm arc} \cos a = - i \mathop{\rm ln} ( a \pm \sqrt {a ^ {2} - 1 } ).$$

The trigonometric functions $\mathop{\rm tan} z$ and $\mathop{\rm cot} z$ take all complex values except $\pm i$: The equations $\mathop{\rm tan} z = a$, $\mathop{\rm cot} z = a$ each have infinitely many solutions for any complex number $a \neq \pm i$:

$$z = \mathop{\rm arc} \mathop{\rm tan} a = \ { \frac{i}{2} } \mathop{\rm ln} \frac{1 - ia }{1 + ia } ,$$

$$z = \mathop{\rm arc} \mathop{\rm cot} a = { \frac{i}{2} } \mathop{\rm ln} \frac{ia + 1 }{ia - 1 } .$$

The trigonometric functions can be expressed in terms of the exponential function:

$$\sin z = { \frac{1}{2i} } ( e ^ {iz} - e ^ {-} iz ),$$

$$\cos z = { \frac{1}{2} } ( e ^ {iz} + e ^ {-} iz ),\ \ \mathop{\rm tan} z = { \frac{1}{i} } \frac{e ^ {iz} - e ^ {-} iz }{e ^ {iz} + e ^ {-} iz } ,$$

and the hyperbolic functions:

$$\sin z = - i \sinh iz,\ \ \cos z = \cosh iz,\ \ \mathop{\rm tan} z = - i \mathop{\rm tanh} iz.$$

The trigonometric functions are also called circular functions.

A formal definition of $\sin z$ and $\cos z$( independent of a picture) can be given by power series and as follows. First of all it can easily be proved that from the previous, visual definition of $\sin x$ and $\cos x$ follows:

$$\mathop{\rm arc} \sin x = \int\limits _ { 0 } ^ { x } \frac{dx}{\sqrt {1- t ^ {2} } } .$$

This can be taken as a formal definition of $\mathop{\rm arc} \sin x$ and the inverse function of $\mathop{\rm arc} \sin x$ can be taken as a formal definition of $\sin x$.

If $z$ is a complex number $x + iy$, with real $x$ and $y$, one can define $e ^ {z} = e ^ {x} ( \cos y + i \sin y )$, and then for complex $z$ define:

$$\sin z = \frac{e ^ {iz} - e ^ {-} iz }{2i} ,\ \ \cos z = \frac{e ^ {iz} + e ^ {-} iz }{2} .$$

#### References

 [a1] T.M. Apostol, "Calculus" , I , Blaisdell (1967) [a2] A.R.F. Verhey, "Complex variables and applications" , McGraw-Hill (1974) [a3] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972)
How to Cite This Entry:
Trigonometric functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_functions&oldid=49035
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article