## Precalculus (6th Edition) Blitzer

The function will be periodic, if it repeats its character after a certain interval of time. That is, if $f\left( x+T \right)=f\left( x \right)$ then it will be a periodic function.
Sine and cosine functions are periodic because they repeat their character after $2\pi$ , or \begin{align} & \sin \left( x+2\pi \right)=\sin x \\ & \cos \left( x+2\pi \right)=\cos x \\ \end{align} So, the sine and cosine functions are periodic. Example: \begin{align} & \sin \left( \frac{\pi }{4}+2\pi \right)=\sin \frac{\pi }{4} \\ & \cos \left( \frac{\pi }{4}+2\pi \right)=\cos \frac{\pi }{4} \\ \end{align}