Chapter 13.5, Problem 1LC

Solution

To sketch: The graph of the function $y=\cos \left(\frac{1}{2}\theta \right)$ in the interval from $0\text{ to }2\pi$ .

Given:

The function is $y=\cos \left(\frac{1}{2}\theta \right)$ .

Concept used:

The cosine curve of the form $y=a\cos \left(b\theta \right)$ where $a\ne 0$ and $b>0$ has an amplitude of $\left(a\right)=\left|a\right|$ , period $\left(T\right)=\frac{2\pi }{b}$ and number of cycles $\left(b\right)$ in the interval from $0\text{ to }2\pi$ . And use the five-point pattern of $\left(\theta \right)$ values which follow a pattern for $\left(y\right)$ -values as $\max ,0,\min ,0\text{ and }\max$ where the curve cuts the $\left(\theta \right)$ -values.

Graph:

In order to sketch the graph of the cosine function $y=\cos \left(\frac{1}{2}\theta \right)$ , evaluate amplitude $\left(a\right)$ , period $\left(T\right)=\frac{2\pi }{b}$ and number of cycles $\left(b\right)$ in the interval from $0\text{ to }2\pi$ .

  $\begin{array}{c}\text{Amplitude}\left(a\right)=\left|1\right|\mathrm{....}\left(\because a=1\right)\\ =1\\ \text{Period}\left(T\right)=\frac{2\pi }{\frac{1}{2}}\mathrm{....}\left(\because b=\frac{1}{2}\right)\\ =4\pi \\ \text{Number of cycles}\left(b\right)=\frac{1}{2}\mathrm{....}\left(\because b=\frac{1}{2}\right)\\ =0.5\end{array}$

Use amplitude $\left|a\right|=1$ to give maximum value of cosine function from start. And the number of cycles is $\left(b\right)=0.5$ , therefore the number of cycles from $0\text{ to }2\pi$ is $0.5$ . Now divide the period into fourths and identify $\left(\theta \right)$ values for the five-point pattern. Therefore,

  $\begin{array}{c}\frac{T}{4}=\frac{4\pi }{4}\\ =\pi \\ \approx 3.14\end{array}$

Hence, the five-point pattern of $\left(\theta \right)$ values is written as $0,\pi ,2\pi ,3\pi \text{ and }4\pi$ , but we have to sketch the graph from the interval from $0\text{ to }2\pi$ therefore use the $\frac{1}{2}$ cycle of the cosine curve to sketch the graph of the curve $y=\cos \left(\frac{1}{2}\theta \right)$ in the interval from $0\text{ to }2\pi$ as shown below.

  

Interpretation:

From the graph of cosine curve $y=\cos \left(\frac{1}{2}\theta \right)$ , it can be interpreted that the three point pattern of $\left(\theta \right)$ values from five-point pattern of $\left(\theta \right)$ values follows a pattern for $\left(y\right)$ -values as $\max ,0,\text{ and }\min$ where the curve cuts the $\left(\theta \right)$ -values in the interval from $0\text{ to }2\pi$ .