Chapter 13.5, Problem 27PPSE

Solution

To identify: The period, range and amplitude of the cosine function $y=3\cos \left(\theta \right)$ in the interval from $0\text{ to }2\pi$ and round the answer to the nearest hundredth.

The period of the cosine function $y=3\cos \left(\theta \right)$ is $2\pi$ , range $y\in \left[-3,3\right]$ and amplitude $\left|a\right|=3$ in the interval from $0\text{ to }2\pi$ .

Given:

The cosine function is $y=3\cos \left(\theta \right)$ .

Concept used:

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

Calculation:

With the help of concept stated above, the amplitude $\left(A\right)$ , the period $\left(T\right)$ , range $\left(y\right)$ and amplitude $\left(a\right)$ of the cosine function $y=3\cos \left(\theta \right)$ is evaluated as:

  $\begin{array}{c}\text{Period}\left(T\right)=\frac{2\pi }{1}\mathrm{....}\left(\because b=1\right)\\ =2\pi \\ \text{Range}\left(y\right)=-\left|3\right|\le y\le \left|3\right|\\ =-3\le y\le 3\\ \text{Amplitude}\left(a\right)=\left|3\right|\mathrm{....}\left(\because a=3\right)\\ =3\end{array}$

Conclusion:

The period of the cosine function $y=3\cos \left(\theta \right)$ is $2\pi$ , range $y\in \left[-3,3\right]$ and amplitude $\left|a\right|=3$ in the interval from $0\text{ to }2\pi$ .