If sin A = 1/`sqrt(5)`, cos B = 3/`sqrt(10)`, A, B being positive acute angles, then what is (A + B) equal to ?

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Explanation

To find the value of \( A + B \) given that \( \sin A = \frac{1}{\sqrt{5}} \) and \( \cos B = \frac{3}{\sqrt{10}} \), we will use the sine addition formula:

\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

### Step 1: Calculate \( \sin A \) and \( \cos B \)

We are given:
\[ \sin A = \frac{1}{\sqrt{5}} \]
\[ \cos B = \frac{3}{\sqrt{10}} \]

### Step 2: Find \( \cos A \)

Using the Pythagorean identity:
\[ \cos A = \sqrt{1 - \sin^2 A} \]
Substituting \( \sin A \):
\[ \cos A = \sqrt{1 - \left(\frac{1}{\sqrt{5}}\right)^2} = \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

### Step 3: Find \( \sin B \)

Using the Pythagorean identity:
\[ \sin B = \sqrt{1 - \cos^2 B} \]
Substituting \( \cos B \):
\[ \sin B = \sqrt{1 - \left(\frac{3}{\sqrt{10}}\right)^2} = \sqrt{1 - \frac{9}{10}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \]

### Step 4: Substitute values into the sine addition formula

Now substituting \( \sin A \), \( \cos B \), \( \cos A \), and \( \sin B \) into the sine addition formula:
\[ \sin(A + B) = \left(\frac{1}{\sqrt{5}}\right) \left(\frac{3}{\sqrt{10}}\right) + \left(\frac{2}{\sqrt{5}}\right) \left(\frac{1}{\sqrt{10}}\right) \]

Calculating each term:
\[ \sin(A + B) = \frac{3}{\sqrt{5} \cdot \sqrt{10}} + \frac{2}{\sqrt{5} \cdot \sqrt{10}} = \frac{3 + 2}{\sqrt{5} \cdot \sqrt{10}} = \frac{5}{\sqrt{5} \cdot \sqrt{10}} \]

### Step 5: Simplify \( \sin(A + B) \)

We can simplify \( \sin(A + B) \):
\[ \sin(A + B) = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find \( A + B \)

Since \( \sin(A + B) = \frac{1}{\sqrt{2}} \), we know that:
\[ A + B = \frac{\pi}{4} \]

Thus, the final answer is:
\[ \boxed{\frac{\pi}{4}} \]