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