## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.3 Trigonometric Equations II - 6.3 Exercises - Page 280: 50

#### Answer

The solution set is $$\{360^\circ n,60^\circ+360^\circ n,180^\circ+360^\circ n,300^\circ+360^\circ n,n\in Z\}$$

#### Work Step by Step

$$\sin\theta-\sin2\theta=0$$ - Recall the identity: $\sin2\theta=2\sin\theta\cos\theta$ and apply it to the given equation: $$\sin\theta-2\sin\theta\cos\theta=0$$ $$\sin\theta(1-2\cos\theta)=0$$ $$\sin\theta=0\hspace{1cm}\text{or}\hspace{1cm}\cos\theta=\frac{1}{2}$$ 1) First, we solve the equation over the interval $[0^\circ,360^\circ)$ - For $\sin\theta=0$, over the interval $[0^\circ, 360^\circ)$, there are two values of $\theta$ where $\sin\theta=0$, which are $\theta=\{0^\circ,180^\circ\}$. - For $\cos\theta=\frac{1}{2}$, over the interval $[0^\circ, 360^\circ)$, there are also two values of $\theta$ where $\cos\theta=\frac{1}{2}$, which are $\theta=\{60^\circ,300^\circ\}$. Therefore, $$\theta=\{0^\circ,60^\circ,180^\circ,300^\circ\}$$ 2) Solve the equation for all solutions Both sine function and cosine function have period $360^\circ$, so we would add $360^\circ$ to all solutions found in part 1) for $\theta$. $$\theta=\{0^\circ+360^\circ n,60^\circ+360^\circ n,180^\circ+360^\circ n,300^\circ+360^\circ n,n\in Z\}$$ $$\theta=\{360^\circ n,60^\circ+360^\circ n,180^\circ+360^\circ n,300^\circ+360^\circ n,n\in Z\}$$ This is also the solution set.

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