Chapter 6.4, Problem 11CFU

To determine

To state: the amplitude and period of the function and then graph that function.

Answer to Problem 11CFU

Amplitude = $\frac{1}{5}$

Period = $8\pi$

Explanation of Solution

Given information:

  $y=-\frac{1}{5}\cos \frac{\theta }{4}$

Calculation:

The equation of sine function is as follows:

  $y=A\cos k\theta$..........(1)

Here $A$ is an amplitude which is an absolute value of the coefficient and $k=\frac{2\pi }{\text{Period}}$ .

Compare the given equation with equation (1) to get,

  $\begin{array}{c}\text{Amplitude}=\left|A\right|\\ =\left|-\frac{1}{5}\right|\\ =\frac{1}{5}\end{array}$

Thus the amplitude of the function is $\frac{1}{5}$ .

Since $k=\frac{1}{4}$ , so the period will be evaluated as follows:

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\\ =\frac{2\pi }{\left(\frac{1}{4}\right)}\left[\text{Put the value of }k\right]\\ =2\pi \left(4\right)\\ =8\pi \end{array}$

Thus the period is $8\pi$ .

The graph of the function for the given amplitude and period is as follows: