Question

The value of sin $\left(2{\sin }^{-1}0.8\right)$ is equal to.

• A. ${\sin }^{-1}1.2$
• B. ${\sin }^{-1}\left(0.96\right)$
• C. ${\sin }^{-1}\left(0.48\right)$
• D. sin ${1.6}^o$

Answer 1

The correct option is B ${\sin }^{-1}\left(0.96\right)$

Let $y=2{\sin }^{-1}0.8$

Take sine both sides,

$\Rightarrow \sin y=\sin \left(2{\sin }^{-1}0.8\right)=2\sin \left({\sin }^{-1}0.8\right)\cos \left({\sin }^{-1}0.8\right)$

$[\because \sin 2\theta =2\sin \theta \cos \theta ]$

$\Rightarrow \sin y=2\left(0.8\right)\cos \left({\cos }^{-1}\sqrt{1-(0.8{)}^2}\right)$

$\Rightarrow \sin y=1.6\cos \left({\cos }^{-1}0.6\right)=1.6\left(0.6\right)=0.96$

$\therefore y={\sin }^{-1}\left(0.96\right)$

Note: $\sin \left({\sin }^{-1}x\right)=x$ if $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$