If `sinx=(1)/(sqrt5).siny=(1)/(sqrt(10))`, where `0 lt x lt (pi)/(2),0 lt y lt (pi)/(2)`, then what is (x+y) equal to?

Options

A. `pi`
B. `pi//2`
C. `pi//4`
D. 0

Answer

To solve the problem where \( \sin x = \frac{1}{\sqrt{5}} \) and \( \sin y = \frac{1}{\sqrt{10}} \), we need to find \( x + y \).

### Step-by-step Solution:

1. **Find \( \cos x \)**:
We know that \( \cos^2 x + \sin^2 x = 1 \). Therefore,
\[ \cos^2 x = 1 - \sin^2 x = 1 - \left(\frac{1}{\sqrt{5}}\right)^2 = 1 - \frac{1}{5} = \frac{4}{5} \]
Taking the square root gives us:
\[ \cos x = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

2. **Find \( \cos y \)**:
Similarly, we calculate \( \cos y \):
\[ \cos^2 y = 1 - \sin^2 y = 1 - \left(\frac{1}{\sqrt{10}}\right)^2 = 1 - \frac{1}{10} = \frac{9}{10} \]
Taking the square root gives us:
\[ \cos y = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \]

3. **Use the sine addition formula**:
We apply the sine addition formula:
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
Substituting the values we found:
\[ \sin(x + y) = \left(\frac{1}{\sqrt{5}}\right) \left(\frac{3}{\sqrt{10}}\right) + \left(\frac{2}{\sqrt{5}}\right) \left(\frac{1}{\sqrt{10}}\right) \]
Simplifying this:
\[ \sin(x + y) = \frac{3}{\sqrt{50}} + \frac{2}{\sqrt{50}} = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

4. **Find \( x + y \)**:
Since \( \sin(x + y) = \frac{1}{\sqrt{2}} \), we can find \( x + y \):
\[ x + y = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
We know that \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \), thus:
\[ x + y = \frac{\pi}{4} \]

### Final Answer:
\[ x + y = \frac{\pi}{4} \]