Question

If \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \), then the value of \( A + B \) will be:

• A. \( \frac{5}{6} \)
• B. 30°
• C. 45°
• D. 60°

Hint:

For the sum of two angles, use the formula \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) to find the result.

Answer 1

The Correct Option is C

We are given: \[ \tan A = \frac{1}{2} \quad \text{and} \quad \tan B = \frac{1}{3} \] We need to find \( A + B \). We will use the formula for the tangent of the sum of two angles: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Substitute the given values for \( \tan A \) and \( \tan B \): \[ \tan(A + B) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left( \frac{1}{2} \times \frac{1}{3} \right)} \] First, simplify the numerator: \[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \] Now, simplify the denominator: \[ 1 - \left( \frac{1}{2} \times \frac{1}{3} \right) = 1 - \frac{1}{6} = \frac{5}{6} \] So, \[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \] The value of \( \tan(A + B) = 1 \) corresponds to an angle of \( 45^\circ \). Thus, \( A + B = 45^\circ \).