Question

In Problems 69-78, solve each equation on the interval$\tan \left(2\theta \right)+2\sin \theta =0$

Short answer

The solution set is$\theta \in \left\{\frac{2\pi }{3},\frac{4\pi }{3}\right\}$

Step-by-step solution

Given information

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

Using the double angle formula and factoring

Using the formula, we get

$$\begin{gathered} \cos \left(2\theta \right)+5\cos \theta +3=0 \\ 2{\cos }^2\theta -1+5\cos \theta +3=0 \\ 2{\cos }^2\theta +5\cos \theta +2=0 \\ (2\cos \theta +1)(\cos \theta +2)=0 \\ 2\cos \theta +1=0 \\ \cos \theta =-\frac{1}{2} \\ \cos \theta +2=0 \\ \cos \theta =-2 \\ \theta =\frac{2\pi }{3} \\ \theta =\frac{4\pi }{3} \end{gathered}$$