Question

If \( \tan(A + B) = 1 \) and \( \tan(A - B) = \frac{1}{\sqrt{3}} \), \( 0^\circ \leq A + B \leq 90^\circ \), find the values of \( A \) and \( B \).

Hint:

To solve trigonometric equations involving sums or differences of angles, use the known values of standard trigonometric functions for specific angles.

Answer 1

We are given that: \[ \tan(A + B) = 1 \quad \text{and} \quad \tan(A - B) = \frac{1}{\sqrt{3}}. \] Step 1: Solve for \( A + B \). We know that \( \tan 45^\circ = 1 \), so from \( \tan(A + B) = 1 \), we have: \[ A + B = 45^\circ. \] Step 2: Solve for \( A - B \). We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), so from \( \tan(A - B) = \frac{1}{\sqrt{3}} \), we have: \[ A - B = 30^\circ. \] Step 3: Solve the system of equations. We now have the system of equations: \[ A + B = 45^\circ \quad \text{and} \quad A - B = 30^\circ. \] Add the two equations: \[ (A + B) + (A - B) = 45^\circ + 30^\circ \quad \implies \quad 2A = 75^\circ \quad \implies \quad A = 37.5^\circ. \] Substitute \( A = 37.5^\circ \) into \( A + B = 45^\circ \): \[ 37.5^\circ + B = 45^\circ \quad \implies \quad B = 7.5^\circ. \]
Conclusion:
The values of \( A \) and \( B \) are \( A = 37.5^\circ \) and \( B = 7.5^\circ \).