Question

If `sin (A - B) = 1/2, cos (A + B) = 1/2; 0 < A + B ≤ 90^circ`. A > B; find ∠A and ∠B.

Answer 1

Given:

`sin(A - B) = 1/2`

`cos(A + B) = 1/2`

0° < A + B ≤ 90° and A > B.

Step-wise calculation:

1. From `cos(A + B) = 1/2` and 0° < A + B ≤ 90°, the only solution in that interval is A + B = 60°.

2. `sin(A - B) = 1/2`. Since A > B, A – B > 0, so take the acute solution A − B = 30° the other solution 150° would make B negative when combined with A + B = 60°.

3. Solve the linear system:

A + B = 60°

A – B = 30°

Add: 2A = 90°

⇒ A = 45°

Subtract: 2B = 30°

⇒ B = 15°

∠A = 45°, ∠B = 15°.