If `sinA=(1)/(sqrt(10))andsinB=(1)/(sqrt(5))`, where A and B are positive acute angles, then `A+B=?`

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Answer

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

### Step-by-Step Solution:

1. **Identify the values of \( \sin A \) and \( \sin B \)**:
\[ \sin A = \frac{1}{\sqrt{10}}, \quad \sin B = \frac{1}{\sqrt{5}} \]

2. **Calculate \( \cos A \) using the Pythagorean identity**:
We know that \( \sin^2 A + \cos^2 A = 1 \). Therefore,
\[ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{1}{\sqrt{10}}\right)^2 = 1 - \frac{1}{10} = \frac{9}{10} \]
Thus,
\[ \cos A = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \]

3. **Calculate \( \cos B \) using the same identity**:
Similarly, for \( B \),
\[ \cos^2 B = 1 - \sin^2 B = 1 - \left(\frac{1}{\sqrt{5}}\right)^2 = 1 - \frac{1}{5} = \frac{4}{5} \]
Thus,
\[ \cos B = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

4. **Use the sine addition formula**:
The sine addition formula states:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Substituting the known values:
\[ \sin(A + B) = \left(\frac{1}{\sqrt{10}}\right) \left(\frac{2}{\sqrt{5}}\right) + \left(\frac{3}{\sqrt{10}}\right) \left(\frac{1}{\sqrt{5}}\right) \]

5. **Calculate each term**:
\[ \sin(A + B) = \frac{2}{\sqrt{10} \cdot \sqrt{5}} + \frac{3}{\sqrt{10} \cdot \sqrt{5}} = \frac{2 + 3}{\sqrt{10} \cdot \sqrt{5}} = \frac{5}{\sqrt{10} \cdot \sqrt{5}} = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

6. **Determine \( A + B \)**:
Since \( \sin(A + B) = \frac{1}{\sqrt{2}} \), we know that:
\[ A + B = \frac{\pi}{4} + n\pi \quad \text{(for any integer } n\text{)} \]
Given that \( A \) and \( B \) are acute angles, we take \( n = 1 \):
\[ A + B = \frac{\pi}{4} + \frac{\pi}{2} = \frac{5\pi}{4} \]

### Final Answer:
\[ A + B = \frac{5\pi}{4} \]