If sin (A – B) = 1/2, cos (A + B) = 1/2, and 0° < A + B ≤ 90°, A > B, then A =

Question: If sin (A – B) = 1/2, cos (A + B) = 1/2, and 0° < A + B ≤ 90°, A > B, then A =

Answer: 

Given:

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

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

3. 0° < A + B ≤ 90° and A > B

Step-by-step solution:

1. Determine A - B:

   sin(A - B) = 1/2

   The angle whose sine is 1/2 is 30° or 150°. Since A > B, A - B must be positive, so:

   A - B = 30°

2. Determine A + B:

   cos(A + B) = 1/2

   The angle whose cosine is 1/2 is 60° or 300°. Given the condition 0° < A + B ≤ 90°, the valid angle is:

   A + B = 60°

3. Solve the equations:

   We have two equations:

   A - B = 30°

   A + B = 60°

   Add these equations to find A:

   (A - B) + (A + B) = 30° + 60°

   2A = 90°

   A = 45°

Thus, A = 45°