Chapter 13.5, Problem 1GP

To determine

To find: All solutions of the given equation.

Answer to Problem 1GP

  $\theta =\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$ .

Explanation of Solution

Given information: Given equation is $\sin 2\theta =\cos \theta$ , where $0\le \theta \le 2\pi$ .

Calculation:

  $\begin{array}{l}\sin 2\theta =\cos \theta \\ \Rightarrow 2\sin \theta \cos \theta =\cos \theta (\because \sin 2\theta =2\sin \theta \cos \theta )\\ \Rightarrow \cos \theta (2\sin \theta -1)=0\\ \Rightarrow \cos \theta =0or2\sin \theta -1=0\\ \Rightarrow \cos \theta =0,\sin \theta =\frac{1}{2}\\ \Rightarrow \theta =\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}\end{array}$

Thus, all solutions of $\sin 2\theta =\cos \theta$ , if $0\le \theta \le 2\pi$ are $\theta =\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$ .