Question

If sin (A + B) = 1 and cos (A – B) = 1, find A and B.

Answer 1

Given:

1. sin (A + B) = 1

2. cos (A – B) = 1

Step-wise calculation:

1) Use sin θ = 1

sin (A + B) = 1

⇒ A + B = 90° + 360°k, ; k ∈ Z

Equivalently, in radians: `(A + B = π/2 + 2πk)`.

2) Use cos θ = 1

cos (A – B) = 1

⇒ A – B = 360°m, ; m ∈ Z

Equivalently, in radians: (A – B = 2πm).

3) Solve the system

Add the two equations:

(A + B) + (A – B) = (90° + 360°k) + 360°m

2A = 90° + 360° (k + m)

A = 45° + 180° (k + m)

Subtract the second from the first:

(A + B) – (A – B) = (90° + 360°k) – 360°m

2B = 90° + 360° (k – m)

B = 45° + 180° (k – m)

A = 45° + 180°n, ; B = 45° + 180°p where n, p ∈ Z and n, p have the same parity.

“Same parity” means both even or both odd; equivalently (n = k + m, ; p = k – m) for some integers (k, m).

A = 45°, ; B = 45°