📂Functions

Definition of Trigonometric Functions

Overview

Trigonometric functions are functions that associate the angles of a right triangle with their trigonometric ratios.

Definition

The trigonometric functions sine and cosine $\sin, \cos : \mathbb{R} \to \mathbb{R}$ are defined as follows.

$$ \sin \theta := {{ y } \over { \sqrt{x^{2} + y^{2}} }} \\ \cos \theta := {{ x } \over { \sqrt{x^{2} + y^{2}} }} $$

Consequently, secant, cosecant, tangent, and cotangent are defined as follows.

$$ \begin{align*} \tan \theta &:= {{ \sin \theta } \over { \cos \theta }} \qquad, \cos \theta \ne 0 \\ \cot \theta &:= {{ \cos \theta } \over { \sin \theta }} \qquad, \sin \theta \ne 0 \\ \sec \theta &:= {{ 1 } \over { \cos \theta }} \qquad, \cos \theta \ne 0 \\ \csc \theta &:= {{ 1 } \over { \sin \theta }} \qquad, \sin \theta \ne 0 \end{align*} $$

Extension to Complex Functions ¹

The trigonometric functions sine and cosine $\sin, \cos : \mathbb{C} \to \mathbb{C}$ are defined as follows.

$$ \sin z := {{ 1 } \over { i2 }} \left( e^{iz} - e^{-iz} \right) \\ \cos z := {{ 1 } \over { 2 }} \left( e^{iz} + e^{-iz} \right) $$

Basic Properties

• [1] Trigonometric functions are $2 \pi$-periodic functions on the real axis.
• [2] The sine function is an odd function, and the cosine function is an even function.

See also

• The relationship between trigonometric functions and hyperbolic functions in complex analysis
• The relationship between trigonometric functions and exponential functions in complex analysis