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Definition of Trigonometric Functions 📂Functions

Definition of Trigonometric Functions

Overview

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

Definition

Trigonometric Ratios

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

sin⁡θ:=yx2+y2cos⁡θ:=xx2+y2 \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.

tan⁡θ:=sin⁡θcos⁡θ,cos⁡θ≠0cot⁡θ:=cos⁡θsin⁡θ,sin⁡θ≠0sec⁡θ:=1cos⁡θ,cos⁡θ≠0csc⁡θ:=1sin⁡θ,sin⁡θ≠0 \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⁡:C→C\sin, \cos : \mathbb{C} \to \mathbb{C} are defined as follows.

sin⁡z:=1i2(eiz−e−iz)cos⁡z:=12(eiz+e−iz) \sin z := {{ 1 } \over { i2 }} \left( e^{iz} - e^{-iz} \right) \\ \cos z := {{ 1 } \over { 2 }} \left( e^{iz} + e^{-iz} \right)

Basic Properties

See also

  1. Osborne (1999). Complex variables and their applications: p28. ↩︎