# What is the period and amplitude for y=cos9x?

Jul 1, 2018

The period is $= \frac{2}{9} \pi$ and the amplitude is $= 1$

#### Explanation:

The period $T$ of a periodic function $f \left(x\right)$ is such that

$f \left(x\right) = f \left(x + T\right)$

Here,

$f \left(x\right) = \cos 9 x$

Therefore,

$f \left(x + T\right) = \cos 9 \left(x + T\right)$

$= \cos \left(9 x + 9 T\right)$

$= \cos 9 x \cos 9 T + \sin 9 x \sin 9 T$

Comparing $f \left(x\right)$ and $f \left(x + T\right)$

$\left\{\begin{matrix}\cos 9 T = 1 \\ \sin 9 t T = 0\end{matrix}\right.$

$\implies$, $9 T = 2 \pi$

$\implies$, $T = \frac{2 \pi}{9}$

The amplitude is $= 1$ as

$- 1 \le \cos x \le 1$

graph{cos(9x) [-1.914, 3.56, -0.897, 1.84]}