Let a = 6, b = 3 and `cos (A -B) = (4)/(5)`
Angle C is equal to

Options

A. `(3pi)/(4)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. none of these

Explanation

To find the angle \( C \) given \( a = 6 \), \( b = 3 \), and \( \cos(A - B) = \frac{4}{5} \), we can follow these steps:

### Step-by-Step Solution:

1. **Use the Cosine Rule**:
We know that \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). We can express \( \cos A \) and \( \cos B \) in terms of sides \( a \) and \( b \) using the Law of Cosines:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]

2. **Express \( \sin A \) and \( \sin B \)**:
Using the Pythagorean identity, we can find \( \sin A \) and \( \sin B \):
\[ \sin A = \sqrt{1 - \cos^2 A}, \quad \sin B = \sqrt{1 - \cos^2 B} \]

3. **Substituting Values**:
For \( a = 6 \) and \( b = 3 \), we can substitute these values into the equations for \( \cos A \) and \( \cos B \).

4. **Find \( \tan \frac{A - B}{2} \)**:
We can use the half-angle identity:
\[ \tan \frac{A - B}{2} = \frac{\sin(A - B)}{1 + \cos(A - B)} \]
Given \( \cos(A - B) = \frac{4}{5} \), we can find \( \sin(A - B) \) using the identity \( \sin^2(A - B) + \cos^2(A - B) = 1 \).

5. **Calculate \( C \)**:
We know that \( A + B + C = 180^\circ \) (or \( \pi \) radians). Therefore, we can find \( C \) as:
\[ C = \pi - (A + B) \]

6. **Final Calculation**:
After calculating \( A \) and \( B \) using the values of \( a \) and \( b \), we can find \( C \).

### Final Answer:
After performing the calculations, we find that \( C = \frac{\pi}{2} \).