Question

If $\sin x=\frac{1}{\sqrt{5}},\sin y=\frac{1}{\sqrt{10}}$, where $0<x<\frac{\pi }{2},0<y<\frac{\pi }{2}$, then what is $(x+y)$ equal to?

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $0$

Answer 1

The correct option is C $\frac{\pi }{4}$

$\sin x=\frac{1}{\sqrt{5}},\sin y=\frac{1}{\sqrt{10}};0<x<\frac{\pi }{2},0<y<\frac{\pi }{2}$

$\therefore x={\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right),y={\sin }^{-1}\left(\frac{1}{\sqrt{10}}\right)$

${\sin }^{-1}\left(a\right)+{\sin }^{-1}\left(b\right)={\sin }^{-1}(a\sqrt{1-b^2}+b\sqrt{1-a^2}$

$={\sin }^{-1}(\frac{1}{\sqrt{5}}\cdot \frac{3}{\sqrt{10}}+\frac{1}{\sqrt{10}}\cdot \frac{2}{\sqrt{5}})$

$={\sin }^{-1}\left(\frac{3+2}{\sqrt{5}\sqrt{10}}\right)={\sin }^{-1}\left(\sqrt{\frac{5}{10}}\right)$

$={\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi }{4}$.