Amplitude and Period of Sine and Cosine Functions

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

Amplitude = $| a |$

Let $b$ be a real number. The period of $y = a \sin (b x)$ and $y = a \cos (b x)$ is given by

$Period = \frac{2 π}{| b |}$

Example:

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

Solution:

Compare the functions

y = \frac{5}{2} \cos (\frac{x}{4})

and

y = a \cos (b x)

to find

a

and

b

$\begin{array}{l} Amplitude = | a | \\ = | \frac{5}{2} | \\ = \frac{5}{2} \end{array}$

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

$\begin{array}{l} Period = \frac{2 π}{| 0.25 |} \\ = 8 π \end{array}$

The period of the function is $8 π$ .