Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 6 - Section 6.2 - More on Trigonometric Equations - 6.2 Problem Set - Page 332: 33



Work Step by Step

$sin(\frac{\theta}{2})-cos(\theta)=0$ $sin(\frac{\theta}{2})-[1-2sin^2(\frac{\theta}{2})]=0$ $sin(\frac{\theta}{2})-1+2sin^2(\frac{\theta}{2})=0$ $2sin^2(\frac{\theta}{2})+sin(\frac{\theta}{2})-1=0$ $sin(\frac{\theta}{2})=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-(1) \pm \sqrt{(1)^2 - 4 (2)(-1)}}{2(2)}=-1,\frac{1}{2}$ $sin(\frac{\theta}{2})=-1$ $\frac{\theta}{2}=sin^{-1}(1)$ $\frac{\theta}{2}=270^o$ $\theta=540^o+270^on\;\;\;$Reduce $sin(\frac{\theta}{2})=\frac{1}{2}$ $\frac{\theta}{2}=sin{-1}(\frac{1}{2})$ We know $sin(\frac{\theta}{2})$ is positive in quadrant $I$ and $II$ $\frac{\theta}{2}=30^o\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\theta=180^o-30^o=150^o$ $\theta=60^o\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\theta=300^o$ Reduce $\theta=\{60^o,300^o\}$
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