Question

If $\frac{\tan 3\theta }{\tan \theta }=4$, then $\frac{\sin 3\theta }{\sin \theta }$ equals:

• A. $\frac{8}{3}$
• B. $\frac{4}{5}$
• C. $\frac{3}{4}$
• D. $Noneofthese$

Answer 1

The correct option is A $\frac{8}{3}$

$\Rightarrow \tan 3\theta =4\tan \theta$

$\Rightarrow \frac{3\tan \theta -{\tan }^3\theta }{1-3{\tan }^2\theta }=4\tan \theta$

$\Rightarrow \frac{3-{\tan }^2\theta }{1-3{\tan }^2\theta }=4$

$\Rightarrow 3-{\tan }^2\theta =4-12{\tan }^2\theta$

$\Rightarrow 11{\tan }^2\theta =1$

$tan\theta =\sqrt{\frac{1}{11}}=\frac{1}{\sqrt{11}}$

$H=\sqrt{1+11}=\sqrt{12}=2\sqrt{3}$

$\sin \theta =\frac{1}{2\sqrt{3}}$

$\Rightarrow \frac{\sin 3\theta }{\sin \theta }=\frac{3\sin \theta -4{\sin }^3\theta }{\sin \theta }$

$\Rightarrow 3-4{\sin }^2\theta$

$\Rightarrow 3-4\times \frac{1}{12}$

$\Rightarrow 3-\frac{1}{3}$

$\Rightarrow \frac{8}{3}$

None of these.