Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 7 - Review Exercises - Page 552: 31


$x=\frac{\pi}{3}+n\pi$ and $x=\frac{2\pi}{3}+n\pi$, where $n$ is an integer.

Work Step by Step

$3~csc^2x=4$ $\frac{3}{4}=\frac{1}{csc^2x}$ $\frac{3}{4}=sin^2x$ $sin~x=±\frac{\sqrt 3}{2}$ The period of $sin~x$ is $2\pi$. The solutions in the interval: $[0,2π)$ are: $x=\frac{\pi}{3}$, $x=\frac{2\pi}{3}$, $x=\frac{4\pi}{3}$ and $x=\frac{5\pi}{3}$ But, $\frac{4\pi}{3}=\frac{\pi}{3}+\pi$ and $\frac{5\pi}{3}=\frac{2\pi}{3}+\pi$ Now, add multiples of $2\pi$ to each of the solutions: $x=\frac{\pi}{3}+n\pi$ and $x=\frac{2\pi}{3}+n\pi$, where $n$ is an integer.
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