Question

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

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

Answer 1

Answer: (D)

Given that, $\sin A=\frac{1}{\sqrt{10}}$ and $\sin B=\frac{1}{\sqrt{5}}$.
We know that,
$\sin (A+B)=\sin A \cos B+\sin B \cos A$
$=\frac{1}{\sqrt{10}} \cdot \sqrt{1-\frac{1}{5}}+\frac{1}{\sqrt{5}} \sqrt{1-\frac{1}{10}}$
$=\frac{1}{\sqrt{10}} \sqrt{\frac{4}{5}}+\frac{1}{\sqrt{5}} \sqrt{\frac{9}{10}}$
$=\frac{1}{\sqrt{50}}(2+3)$
$=\frac{5}{\sqrt{50}}$
$=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin =(A+B)=\sin \frac{\pi}{4}$
$\Rightarrow A+B=\frac{\pi}{4}$