Periodic function

A function $f$ is said to be periodic if it satisfies with constant value $c$

$\displaystyle f(x + c) = f(x)$   $\displaystyle \mbox{ for every $x$. }$ (2-1)

The least such positive value $ c = c_0$ for which (2-1) holds is called the period of $f$. Given the period $ c_0$ of $f$, we have for any integer $n$,

$\displaystyle f(x + c_0 n) = f(x)$   $\displaystyle \mbox{ for every $x$. }$ (2-2)

Period. The sine function and the cosine function are periodic, and both have the period of $ 2 \pi$. That is,

$\displaystyle \sin(x + 2\pi) = \sin x$    and $\displaystyle \quad \cos(x + 2\pi) = \cos x$ (2-3)

(2-2) and (2-3) together implies that we have for any integer $n$,

$\displaystyle \sin(x + 2\pi n) = \sin x$    and $\displaystyle \quad
\cos(x + 2\pi n) = \cos x

Phase shift. The constant value $ \pi$ is not the period for the sine and the cosine functions, but gives the following formulas of phase shift:

$\displaystyle \sin(x + \pi) = -\sin x$    and $\displaystyle \quad
\cos(x + \pi) = -\cos x

Values of sine and cosine functions. Use the ``reference angle $ \theta_R$'' (the acute angle between the terminal side of the original angle $ \theta$ and the $x$-axis) to find the coordinate  $ (\cos\theta, \sin\theta)$ on the unit circle.

Department of Mathematics
Last modified: 2005-09-29