Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - 7.6 Trigonometric Equations - 7.6 Exercises: 7

Answer

$\displaystyle \{\frac{\pi}{6}, \ \ \frac{5\pi}{6}, \ \ \frac{7\pi}{6} , \ \ \frac{11\pi}{6}\}$.

Work Step by Step

As solved in exercise 1, two angles (radian measures) with cosine equal to $\displaystyle \frac{1}{2}$ are $\displaystyle \{\frac{\pi}{3}, \ \ \frac{5\pi}{3}\}$. Also, if $0 \leq x < 2\pi\qquad/\times 2$ then $0 \leq 2x < 4\pi.$ So either $2x= \displaystyle \frac{\pi}{3}\quad$or$\quad 2x= \displaystyle \frac{\pi}{3}+2\pi$ or $2x= \displaystyle \frac{5\pi}{3}\quad$or$\quad 2x= \displaystyle \frac{5\pi}{3}+2\pi$ $... $divide both sides by 2 in each equation $x= \displaystyle \frac{\pi}{6}\quad$or$\quad x= \displaystyle \frac{\pi}{6}$+$\displaystyle \pi=\frac{7\pi}{6}$ or $ x=\displaystyle \frac{5\pi}{6}\quad$or$\quad x= \displaystyle \frac{5\pi}{6}$+$\displaystyle \pi=\frac{11\pi}{6}$ Solution set: $\displaystyle \{\frac{\pi}{6}, \ \ \frac{5\pi}{6}, \ \ \frac{7\pi}{6} , \ \ \frac{11\pi}{6}\}$.
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