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:

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

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{4}$

Explanation for the correct option.

Step 1: Find $\cos A,\cos B$

Given that, $\sin A=\frac{1}{\sqrt{10}}$ and $\sin B=\frac{1}{\sqrt{5}}$.

We know that, ${\sin }^2\theta +{\cos }^2\theta =1$.

$\begin{array}{rcl}\cos A& =& \sqrt{1-{\sin }^{2}A}\\ & =& \sqrt{1-{\left(\frac{1}{\sqrt{10}}\right)}^2}\\ & =& \frac{3}{\sqrt{10}}\end{array}$

Similarly, $\cos B=\frac{2}{\sqrt{5}}$

Step 2: Find the value of $A+B$

$\begin{array}{rcl}\sin \left(A+B\right)& =& \sin A\cos B+\cos A\sin B\\ & =& \frac{1}{\sqrt{10}}·\frac{2}{\sqrt{5}}+\frac{3}{\sqrt{10}}·\frac{1}{\sqrt{5}}\\ & =& \frac{2+3}{\sqrt{50}}\\ & =& \frac{5}{5\sqrt{2}}\\ & =& \frac{1}{\sqrt{2}}\end{array}$

$\Rightarrow A+B=\frac{\mathrm{\pi }}{4}$

Hence, option D is correct.