Chapter 13.7, Problem 13PPS

To determine

To determine the amplitude and period of the given function and graph it.

Answer to Problem 13PPS

Amplitude $=1$ , Period $=\frac{2\pi }{3}$

Explanation of Solution

Given:

  $y=\cos \frac{1}{2}\theta$

Concept used:

Amplitude is the distance between the center line of the function and the top or bottom of the function.

Period is the distance between two peaks of the graph.

The generalized equation for a cosine function is:

  $y=A\cos \left(Bx\right)$

Amplitude $=A$

Period $=\frac{2\pi }{B}$

Calculation:

For $y=\cos \frac{1}{2}\theta$ ,

Comparing with the general cosine equation $y=A\cos \left(Bx\right)$ ,

  $A=1,B=\frac{1}{2}$

So, the amplitude of the given function is $1$ .

Period $=\frac{2\pi }{B}$

  $\begin{array}{l}=\frac{2\pi }{\frac{1}{2}}\\ =2\pi \times 2\\ =4\pi \end{array}$

The period of the given function is $4\pi$ .

Graph:

The graph of $y=\cos \frac{1}{2}\theta$:

  

Conclusion:

Therefore, the amplitude of the given function is $1$ and the period is $4\pi$ .