Precalculus (6th Edition) Blitzer

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

Chapter 5 - Section 5.5 - Trigonometric Equations - Exercise Set - Page 704: 99


The solution is $\underline{x=0,\frac{2\pi }{3},\pi ,\frac{4\pi }{3}}.$

Work Step by Step

We have to solve $\sin 2x+\sin x=0$: $\begin{align} & \sin 2x+\sin x=0 \\ & 2\sin x\cos x+\sin x=0 \end{align}$ Factor: $\sin x\left( 2\cos x+1 \right)=0$ So, from the above result, it can be concluded that, $\sin x=0$ And the other value will be: $2\cos x+1=0$ With the general formula: $\begin{align} & \sin x=0 \\ & x=n\pi \end{align}$ Compute the value of $\cos x$. $\begin{align} & 2\cos x+1=0 \\ & 2\cos x=-1 \\ & \cos x=\frac{-1}{2} \end{align}$ So, in the interval $\left[ 0,2\pi \right),$ the cosine function is $\frac{-1}{2}$ at $\frac{2\pi }{3}\text{ and }\frac{4\pi }{3}$ according to the trigonometric table. $\begin{align} & \cos x=\frac{-1}{2} \\ & x=\frac{2\pi }{3} \end{align}$ The other value is: $\begin{align} & \cos x=\frac{-1}{2} \\ & x=\frac{4\pi }{3} \end{align}$ Since the period of the cosine function is $2\pi $, the general solution of the equation is: $x=\frac{2\pi }{3}+2n\pi \,\text{ and }\frac{4\pi }{3}+2n\pi $ To get different solutions, $\text{put }n=0,1,2,3\ldots $ $\begin{align} & \text{For }n=0, \\ & x=n\pi \\ & =0 \end{align}$ So, another value is: $\begin{align} & x=\frac{2\pi }{3}+2n\pi \\ & =\frac{2\pi }{3} \end{align}$ Other value of x is: $\begin{align} & x=\frac{4\pi }{3}+2n\pi \\ & =\frac{4\pi }{3} \end{align}$ Substitute the value of n: $\begin{align} & \text{For }n=1, \\ & x=n\pi \\ & =\pi \end{align}$ When putting the other values of $n$, the solution becomes outside the interval $\left[ 0,2\pi \right).$ Thus, the exact solution of the equation $\sin 2x+\sin x=0$ on the interval $\left[ 0,2\pi \right)$ is $\underline{x=0,\frac{2\pi }{3},\pi ,\frac{4\pi }{3}}.$
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.