# Amplitude and Period of Sine and Cosine Functions

The amplitude of $y=a\mathrm{sin}\left(x\right)$ and $y=a\mathrm{cos}\left(x\right)$ represents half the distance between the maximum and minimum values of the function.

Amplitude = $|a|$

Let $b$ be a real number. The period of $y=a\mathrm{sin}\left(bx\right)$ and $y=a\mathrm{cos}\left(bx\right)$ is given by

$\text{Period}=\frac{2\pi }{|b|}$

Example:

Find the period and amplitude of $y=\frac{5}{2}\mathrm{cos}\left(\frac{x}{4}\right)$ .

Solution:

Compare the functions $y=\frac{5}{2}\mathrm{cos}\left(\frac{x}{4}\right)$ and $y=a\mathrm{cos}\left(bx\right)$ to find $a$ and $b$ .

$\begin{array}{l}\text{Amplitude}=|a|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=|\frac{5}{2}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{5}{2}\end{array}$

The amplitude of the function is $\frac{5}{2}$ .

$\begin{array}{l}\text{Period}=\frac{2\pi }{|0.25|}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=8\pi \end{array}$

The period of the function is $8\pi$ .