Question

If \( \tan(A + B) = \sqrt{3} \) and \( \cos(A - B) = \frac{\sqrt{3}}{2} \), the values of \( A \) and \( B \) are:

• A. \( 40^\circ, 20^\circ \)
• B. \( 15^\circ, 30^\circ \)
• C. \( 45^\circ, 15^\circ \)
• D. \( 60^\circ, 30^\circ \)

Hint:

When given trigonometric identities involving sums and differences of angles, use standard values from trigonometric tables or the unit circle to find the angles.

Answer 1

The Correct Option is C

We are given that \( \tan(A + B) = \sqrt{3} \), which implies: \[ A + B = 60^\circ \quad \text{(since \( \tan 60^\circ = \sqrt{3} \))} \] Also, \( \cos(A - B) = \frac{\sqrt{3}}{2} \), which implies: \[ A - B = 30^\circ \quad \text{(since \( \cos 30^\circ = \frac{\sqrt{3}}{2} \))} \] Now, solving the system of equations: \[ A + B = 60^\circ \] \[ A - B = 30^\circ \] Adding these two equations: \[ 2A = 90^\circ \quad \Rightarrow \quad A = 45^\circ \] Substituting \( A = 45^\circ \) into \( A + B = 60^\circ \): \[ 45^\circ + B = 60^\circ \quad \Rightarrow \quad B = 15^\circ \] Therefore, the values of \( A \) and \( B \) are \( 45^\circ \) and \( 15^\circ \).