Question

If $\sin A=\frac{1}{\sqrt{10}}$ and $\sin B=\frac{1}{\sqrt{5}}$where $A$ and $B$ are positive acute angles, then $A+B=$

• A. $\pi$
• B. $\pi /2$
• C. $\pi /3$
• D. $\pi /4$

Answer 1

The correct option is D $\pi /4$

$\sin A=\frac{1}{\sqrt{10}}\Rightarrow A={\sin }^{-1}\left(\frac{1}{\sqrt{10}}\right)$

$\sin B=\frac{1}{\sqrt{5}}\Rightarrow B={\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right)$

Now,

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

As we know that,

${\sin }^{-1}A+{\sin }^{-1}B={\sin }^{-1}(A\sqrt{1-B^2}+B\sqrt{1-A^2})$

$\therefore A+B={\sin }^{-1}(\frac{1}{\sqrt{10}}\sqrt{1-\frac{1}{5}}+\frac{1}{\sqrt{5}}\sqrt{1-\frac{1}{10}})$

$A+B={\sin }^{-1}(\frac{2}{\sqrt{50}}+\frac{3}{\sqrt{50}})$

$\Rightarrow A+B={\sin }^{-1}\left(\frac{5}{\sqrt{50}}\right)$

$\Rightarrow A+B={\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\Rightarrow A+B=\frac{\pi }{4}(\because \text{A and B are positive acute angles})$

Hence the correct answer is $\frac{\pi }{4}$.