Question

If tan A = cot (A + 10°), where (A + 10°) is acute then find ∠A.

Answer 1

Given: tan A = cot(A + 10°) and (A + 10°) is acute.

Step-wise calculation:

1. Use cot x = tan(90° – x):

cot(A + 10°) = tan(90° – (A + 10°))

= tan(80° – A)

2. So tan A = tan(80° – A)

⇒ A = 80° – A + k·180°, k ∈ Z.

3. Solve: 2A = 80° + k·180°

⇒ A = 40° + k·90°

4. Since (A + 10°) is acute:

0° < A + 10° < 90°

⇒ –10° < A < 80°

5. Test k values:

k = 0 → A = 40°   ...(Valid)

k = 1 → A = 130°   ...(Invalid)

k = –1 → A = –50°   ...(Invalid)

A = 40°.