Chapter 10, Problem 7CR

To determine

The solution set of the equation $\sin \left(2\theta \right)=0.5$

Answer to Problem 7CR

Solution:

The solution set of the equation $\sin \left(2\theta \right)=0.5$ is $\left\{\frac{\pi }{12}+k\pi ,\frac{5\pi }{12}+k\pi \right\}$, where $k\in \mathbb{Z}$

Explanation of Solution

Given information:

The equation $\sin \left(2\theta \right)=0.5$

To find the solution of the equation $\sin \left(2\theta \right)=0.5$, as follows:

By applying ${\sin }^{-1}$ on both sides of the equation,

  $\Rightarrow 2\theta ={\sin }^{-1}\left(0.5\right)$

  $\Rightarrow 2\theta =\frac{\pi }{6}+2k\pi$ and $2\theta =\frac{5\pi }{6}+2k\pi$, where $k\in \mathbb{Z}$

  $\Rightarrow \theta =\frac{\pi }{12}+k\pi$ and $\theta =\frac{5\pi }{12}+k\pi$, where $k\in \mathbb{Z}$

The solution set is $\left\{\frac{\pi }{12}+k\pi ,\frac{5\pi }{12}+k\pi \right\}$

Therefore, the solution of the equation $\sin \left(2\theta \right)=0.5$ is $\left\{\frac{\pi }{12}+k\pi ,\frac{5\pi }{12}+k\pi \right\}$, where $k\in \mathbb{Z}$ .