Question

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

Hint:

Memorize key tangent values: \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), \( \tan 45^\circ = 1 \), \( \tan 60^\circ = \sqrt{3} \). They help solve trig equations quickly.

Answer 1

Concept: We use standard tangent values:

• \( \tan 60^\circ = \sqrt{3} \)
• \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)

Thus, we convert the trigonometric equations into angle equations and solve simultaneously.
Step 1: Use known tangent values.
\[ \tan(A + B) = \sqrt{3} \Rightarrow A + B = 60^\circ \quad (\text{principal value}) \] \[ \tan(A - B) = \frac{1}{\sqrt{3}} \Rightarrow A - B = 30^\circ \]
Step 2: Solve the system of equations.
Add both equations: \[ (A + B) + (A - B) = 60^\circ + 30^\circ \] \[ 2A = 90^\circ \Rightarrow A = 45^\circ. \]
Step 3: Substitute \( A \) back.
\[ A + B = 60^\circ \] \[ 45^\circ + B = 60^\circ \Rightarrow B = 15^\circ. \] Conclusion:
The required angles are: \[ A = 45^\circ, \quad B = 15^\circ. \]