Algebra 2 (1st Edition)

Published by McDougal Littell
ISBN 10: 0618595414
ISBN 13: 978-0-61859-541-9

Chapter 14 Trigonometric Graphs, Identities, and Equations - 14.4 Solve Trigonometric Equations - Guided Practice for Examples 4, 5, and 6 - Page 934: 5


$$x=2\pi n,\:x=\frac{2\pi }{3}+2\pi n$$

Work Step by Step

We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving this, we find: $$\left(1-\cos \left(x\right)\right)^2=\left(\sqrt{3}\sin \left(x\right)\right)^2 \\ \left(1-\cos \left(x\right)\right)^2-3\sin ^2\left(x\right)=0 \\ \left(1-\cos \left(x\right)\right)^2-\left(1-\cos ^2\left(x\right)\right)\cdot \:3=0 \\ \cos \left(x\right)=1,\:\cos \left(x\right)=-\frac{1}{2} \\ x=2\pi n,\:x=\frac{2\pi }{3}+2\pi n$$
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