Chapter 13.5, Problem 17PPSE

Solution

To write: A cosine function whose amplitude is $\frac{\pi }{2}$ and period $3$ assuming $a>0$ .

The equation of cosine curve is written as $y=\frac{\pi }{2}\cos \left(\frac{2\pi }{3}\theta \right)$ where $a>0$ .

Given:

The amplitude $\left(a\right)$ of cosine curve is $\frac{\pi }{2}$ and period $3$ .

Concept used:

The cosine curve is of the form $y=a\cos \left(b\theta \right)$ where $a\ne 0$ and $b>0$ has an amplitude of $\left(a\right)=\left|a\right|$ , period $\left(T\right)=\frac{2\pi }{b}$ and number of cycles $\left(b\right)$ in the interval from $0\text{ to }2\pi$ .

Calculation:

In order to write the equation of cosine function, evaluate the missing parameters using the concept stated above.

As $\left|a\right|=\frac{\pi }{2}$ and period $\left(T\right)=3$ . Therefore, the maximum and minimum value of the function is $\frac{\pi }{2}\text{ and }-\frac{\pi }{2}$ respectively.

  $\begin{array}{c}\text{And, period}\left(T\right)=3\text{ gives,}\\ \frac{2\pi }{b}=3\\ b=\frac{2\pi }{3}\end{array}$

Since, the number of cycles $\left(b\right)=\frac{2\pi }{3}$ and amplitude $\left|a\right|=\frac{\pi }{2}$ . Therefore, the equation of cosine curve of the form $y=a\cos \left(b\theta \right)$ is written as $y=\frac{\pi }{2}\cos \left(\frac{2\pi }{3}\theta \right)$ .

Conclusion:

The equation of cosine curve is written as $y=\frac{\pi }{2}\cos \left(\frac{2\pi }{3}\theta \right)$ where $a>0$ .