Question

In Problems $69-78$, solve each equation on the interval $0\le \theta <2\pi$

$\tan \left(2\theta \right)+2\sin \theta =0$

Short answer

The solution set is$\left\{0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}\right\}$

Step-by-step solution

Given information

Given function$\tan \left(2\theta \right)+2\sin \theta =0$

rewrite the expression in terms of sine and cosine

Rewriting, we get

$$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ \frac{\sin 2\theta }{\cos 2\theta }+2\sin \theta =0 \\ \sin 2\theta +2\sin \theta \cos 2\theta =0 \\ 2\sin \theta \cos \theta +2\sin \theta \cos 2\theta =0 \\ 2\sin \theta (\cos \theta +\cos 2\theta )=0 \end{gathered}$$

further solving using identities

Further solving, we get

$$\begin{gathered} 2\sin \theta =0 \\ \sin \theta =0 \\ \theta =0,\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} \cos \theta +2{\cos }^2\theta -1=0 \\ (\cos \theta +1)(2\cos \theta -1)=0 \\ \cos \theta +1=0 \\ 2\cos \theta -1=0 \\ 2\cos \theta =1 \\ \cos \theta =\frac{1}{2} \\ \theta =\left\{0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}\right\} \end{gathered}$$