Question

Find the measure of angles A and B in the following, where A and B are acute.

2 sin (2A − B) = 1 and tan `((2A + B)/2) = 1/sqrt3`

Answer 1

Given:

2 sin (2A − B) = 1 and tan `((2A + B)/2) = 1/sqrt3`

Step 1: 

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

2A − B = 30°    ... [since A and B are acute, we only consider this solution.]

Step 2:

tan `(2A + B/2) = 1/sqrt3 = (2A +  B)/2 = 30° = 2A + B = 60°`

Step 3:

2A − B = 30°

2A + B = 60°

(2A − B) + (2A + B) = 30° + 60° ⇒ 4A = 90° ⇒ A = 22.5°    ... [Add both equations.]

A  = 22.5° into  2A − B = 30° 

2(22.5°) − B = 30° ⇒ 45° − B = 30° ⇒ B = 15°

A = 22.5°, B = 15°   ...[both angles A and B are acute]