Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.3 Trigonometric Equations II - 6.3 Exercises - Page 280: 49


The solution set is $$\{30^\circ+180^\circ n,90^\circ+180^\circ n,150^\circ+180^\circ n,n\in Z\}$$

Work Step by Step

$$2\cos^22\theta=1-\cos2\theta$$ $$2\cos^22\theta+\cos2\theta-1=0$$ $$(2\cos^22\theta+2\cos2\theta)+(-\cos2\theta-1)=0$$ $$2\cos2\theta(\cos2\theta+1)-(\cos2\theta+1)=0$$ $$(\cos2\theta+1)(2\cos2\theta-1)=0$$ $$\cos2\theta=-1\hspace{1cm}\text{or}\hspace{1cm}\cos2\theta=\frac{1}{2}$$ 1) First, we solve the equation over the interval $[0^\circ,360^\circ)$ - For $\cos2\theta=-1$, over the interval $[0^\circ, 360^\circ)$, there is only one value of $\theta$ where $\cos2\theta=-1$, which is $2\theta=\{180^\circ\}$. - For $\cos2\theta=\frac{1}{2}$, over the interval $[0^\circ, 360^\circ)$, there are two values of $\theta$ where $\cos2\theta=\frac{1}{2}$, which are $2\theta=\{60^\circ,300^\circ\}$. Therefore, $$2\theta=\{60^\circ,180^\circ,300^\circ\}$$ 2) Solve the equation for all solutions Cosine function has period $360^\circ$, so we would add $360^\circ$ to all solutions found in part 1) for $2\theta$. $$2\theta=\{60^\circ+360^\circ n,180^\circ+360^\circ n,300^\circ+360^\circ n,n\in Z\}$$ Finally, we find the solutions for $\theta$ by dividing all the solutions for $2\theta$ by $2$, which is also the solution set: $$\theta=\{30^\circ+180^\circ n,90^\circ+180^\circ n,150^\circ+180^\circ n,n\in Z\}$$
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