Question

For acute angles A and B and A + 2B and 2A + B are acute if `tan (A + 2B) = sqrt(3)` and `sin (2A + B) = 1/sqrt(2)`, then find the measures of angles A and B.

Answer 1

From `tan (A + 2B) = sqrt(3)`:

Since `60^circ = sqrt(3)`, we have:

A + 2B = 60°   ...(1)

From `sin (2A + B) = 1/sqrt(2)`:

Since `sin 45^circ = 1/sqrt(2)`, we have:

2A + B = 45°   ...(2)

Solving the equations:

Multiply equation (2) by 2:

4A + 2B = 90°   ...(3)

Subtract equation (1) from (3):

(4A + 2B) – (A + 2B) = 90° – 60°

3A = 30°

⇒ A = 10°

Substitute A = 10° in equation (2):

2(10°) + B = 45°

20° + B = 45°

⇒ B = 25°

The measures of the angles are A = 10° and B = 25°.