Question

If $\sin A=\frac{2}{\sqrt{5}}$ and $\cos B=\frac{1}{\sqrt{10}}$ where $A$ and $B$ are acute angles, then what is $A+B$ equal to?

• A. ${135}^o$
• B. ${90}^o$
• C. ${75}^o$
• D. ${60}^o$

Answer 1

The correct option is A ${135}^o$

$\sin A=\frac{2}{\sqrt{5}}\Rightarrow \cos A=\frac{1}{\sqrt{5}}\cos B=\frac{1}{\sqrt{10}}\sin B=\frac{3}{\sqrt{10}}$

Then, $\cos (A+B)=\frac{1}{\sqrt{5}}\frac{1}{\sqrt{10}}-\frac{2}{\sqrt{5}}\frac{3}{\sqrt{10}}$

$=\frac{1}{\sqrt{50}}-\frac{6}{\sqrt{50}}$

$=-\frac{5}{\sqrt{50}}$

$=-\frac{1}{\sqrt{2}}$

Thus, $A+B={135}^o$