Question

If $\sin^{-1} \alpha = \tan^{-1} \frac{3}{4}$, then $\alpha $ equals :

• A. $\frac{3}{5}$
• B. 1
• C. $\frac{2}{5}$
• D. $\frac{3}{4}$

Answer 1

Answer: (A)

Let $\sin^{-1} \alpha = \tan^{-1} \frac{3}{4}$ .......(1)
To find $\alpha$
Put $ \tan^{-1} \frac{3}{4} = \theta \Rightarrow \tan\theta = \frac{3}{4} $
Now equation (i) becomes
$\sin^{-1} \alpha = \theta \Rightarrow \alpha = \sin\theta $
$\Rightarrow \alpha = \frac{ 1}{cosec \theta} = \frac{ 1}{\sqrt{1+ \cot^{2} \theta}}$
$\Rightarrow \alpha = \frac{1}{\sqrt{1+\frac{1}{\tan^{2} \theta}} } = \frac{ 1}{\sqrt{1 + \left(\frac{4}{3}\right)^{2}} } $
$\Rightarrow \alpha = \frac{3}{5} $