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

$Solution:$ $$\begin{array}{l}\text { Since, } \sin A=\frac{1}{\sqrt{10}} \text { and } \sin B=\frac{1}{\sqrt{5}} \\\Rightarrow \quad \cos A=\frac{3}{\sqrt{10}} \text { and } \cos B=\frac{2}{\sqrt{5}} \\\therefore \quad \cos (A+B)=\cos A \cos B-\sin A \sin B \\=\frac{3}{\sqrt{10}} \times \frac{2}{\sqrt{5}}-\frac{1}{\sqrt{10}} \times \frac{1}{\sqrt{5}}=\frac{1}{\sqrt{2}} \\\Rightarrow \quad A+B=\frac{\pi}{4} \\\end{array}$$

$Correct$ $Option$ $is$ ($d$).