Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 660: 99

Answer

Amplitude is $3$ and the period is $4\pi $

Work Step by Step

The given equation is $y=3\sin \frac{1}{2}x$. It is in the form of $y=A\operatorname{sinBx}$, A is 3 and B is $\frac{1}{2}$. It shows the amplitude of 3 and the period is, $\begin{align} & \frac{2\pi }{B}=\frac{2\pi }{\frac{1}{2}} \\ & =2\pi \times 2 \\ & =4\pi \end{align}$ The quarter period is, $\frac{4\pi }{4}=\pi $ Since it is mentioned that the cycles begin from 0, we add the quarter period in order to generate the values of x. Find the various values of the given function as follows: When the value of x is 0: $\begin{align} & y=3\sin \left( \frac{1}{2}x \right) \\ & =3\sin \left( \frac{1}{2}\times 0 \right) \\ & =0 \end{align}$ And the coordinates of the graph are: $\left( 0,0 \right)$. When the value of x is $\pi $: $\begin{align} & y=3\sin \left( \frac{1}{2}x \right) \\ & =3\sin \left( \frac{1}{2}\times \pi \right) \\ & =3\sin \left( \frac{\pi }{2} \right) \end{align}$ And the value at $\frac{\pi }{2}$ is 1. So, $\begin{align} & y=3\times 1 \\ & =3 \end{align}$ And the coordinates of the graph are $\left( \pi,3 \right)$. When the value of x is $2\pi $: $\begin{align} & y=3\sin \left( \frac{1}{2}x \right) \\ & =3\sin \left( \frac{1}{2}\times 2\pi \right) \\ & =3\sin \left( \frac{2\pi }{2} \right) \end{align}$ The value at $\pi $ is 0. Then, $\begin{align} & y=3\times 0 \\ & =0 \end{align}$ And the coordinates of the graph are $\left( 2\pi,0 \right)$. When the value of x is $3\pi $. So, $\begin{align} & y=3\sin \left( \frac{1}{2}x \right) \\ & =3\sin \left( \frac{1}{2}\times 3\pi \right) \\ & =3\sin \left( \frac{3\pi }{2} \right) \end{align}$ The value at $\frac{3\pi }{2}$ is -1 $\begin{align} & y=3\times -1 \\ & =-3 \end{align}$ And the coordinates of the graph are $\left( 3\pi,-3 \right)$. When the value of x is $4\pi $. So, $\begin{align} & y=3\sin \left( \frac{1}{2}x \right) \\ & =3\sin \left( \frac{1}{2}\times 4\pi \right) \\ & =3\sin \left( \frac{4\pi }{2} \right) \end{align}$ The value of $2\pi $ is 0, $\begin{align} & y=3\times 0 \\ & =0 \end{align}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.