# If x and y are acute such that

Question:

If x and y are acute such that $\sin x=\frac{1}{\sqrt{5}}$ and $\sin y=\frac{1}{\sqrt{10}}$ prove that $(x+y)=\frac{\pi}{4}$

Solution:

Givensin $x=\frac{1}{\sqrt{5}}$ andsiny $=\frac{1}{\sqrt{10}}$

Now we will calculate value of cos x and cosy

$\cos x=\sqrt{\left(1-\sin ^{2} x\right)} \Rightarrow \sqrt{\left(1-\left(\frac{1}{\sqrt{5}}\right)^{2}\right)}=\sqrt{\left(\frac{5-1}{5}\right)} \Rightarrow \sqrt{\left(\frac{4}{5}\right)}=\frac{2}{\sqrt{5}}$

$\cos y=\sqrt{\left(1-\sin x^{2}\right)} \Rightarrow \sqrt{\left(1-\left(\frac{1}{\sqrt{10}}\right)^{2}\right)}=\sqrt{\left(\frac{10-1}{10}\right)} \Rightarrow \sqrt{\left(\frac{9}{10}\right)}=\frac{3}{\sqrt{10}}$

$\operatorname{Sin}(x+y)=\sin x \cdot \cos y+\cos x \cdot \sin y$

$=\frac{1}{\sqrt{5}} \cdot \frac{3}{\sqrt{10}}+\frac{2}{\sqrt{5}} \cdot \frac{1}{\sqrt{10}} \Rightarrow \frac{3+2}{\sqrt{50}}=\frac{5}{5 \sqrt{2}} \Rightarrow \frac{1}{\sqrt{2}}$

$\Rightarrow \sin (\mathrm{x}+\mathrm{y})=\frac{1}{\sqrt{2}}$

$\Rightarrow \mathrm{x}+\mathrm{y}=\frac{\pi}{4}$