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Elementary Functions > Csc[z] > Introduction to the trigonometric functions

The best-known properties and formulas for trigonometric functions

Real values for real arguments

For real values of argument , the values of all the trigonometric functions are real (or infinity).

In the points , the values of trigonometric functions are algebraic. In several cases they can even be rational numbers or integers (like or ). The values of trigonometric functions can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

Simple values at zero

All trigonometric functions have rather simple values for arguments and :


All trigonometric functions are defined for all complex values of , and they are analytical functions of over the whole complex ‐plane and do not have branch cuts or branch points. The two functions and are entire functions with an essential singular point at . All other trigonometric functions are meromorphic functions with simple poles at points for and , and at points for and .


All trigonometric functions are periodic functions with a real period ( or ):

Parity and symmetry

All trigonometric functions have parity (either odd or even) and mirror symmetry:

Simple representations of derivatives

The derivatives of all trigonometric functions have simple representations that can be expressed through other trigonometric functions:

Simple differential equations

The solutions of the simplest second‐order linear ordinary differential equation with constant coefficients can be represented through and :

All six trigonometric functions satisfy first-order nonlinear differential equations: