Question

If $\frac{1}{2}{\sin }^{-1}\left(\frac{3\sin \theta }{5+4\cos \theta }\right)=\frac{\pi }{4}$, then $4\tan \theta$ is equal to

• A. $1/3$
• B. $-3$
• C. $1$
• D. $2$

Answer 1

The correct option is B $-3$

$\frac{1}{2}{\text{sin}}^{-1}(\frac{3\sin \theta }{5+4\cos \theta })=\frac{\pi }{4}$

$⟹\frac{3\sin \theta }{5+4\cos \theta }=\sin \frac{2\pi }{4}$

$⟹3\sin \theta =5+4\cos \theta$

$⟹3\tan \theta =5\text{sec}\theta +4$

$⟹(3\tan \theta -4)=5\text{sec}\theta$

$⟹(3\tan \theta -4{)}^2=(5\text{sec}\theta {)}^2$

$⟹9{\tan }^2\theta -24\tan \theta +16=25(1+{\tan }^2\theta )$

$⟹16{\tan }^2\theta +24\tan \theta +9=0$

$⟹(4\tan \theta +3{)}^2=0$

$⟹4\tan \theta =-3$