Question

If cos 2A = sin (A – 15°), where 2A is acute then find ∠A.

Answer 1

Given: cos 2A = sin (A – 15°), where 2A is acute (so 0° < A < 45°).

Step-wise calculation:

1. Write sin in terms of cos:

sin(A – 15°) = cos[90° – (A – 15°)]

= cos(105° – A)

2. So cos 2A = cos(105° – A).

3. cos x = cos y

⇒ x = ±y + 360°k.

For k = 0 we get:

2A = 105° – A

⇒ 3A = 105°

⇒ A = 35°

2A = –(105° – A)

⇒ 2A = A – 105°

⇒ A = –105°   ...(Reject, not in 0° –45°)

4. Other k give angles outside 0° < A < 45°, so the only valid solution is A = 35°.

∠A = 35°.