|
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, …}.
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  ):
All trigonometric functions have parity (either odd or even) and mirror symmetry:
The derivatives of all trigonometric functions have simple representations that can be expressed through other trigonometric functions:
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:
|
