Question

If `sin (A - B) = cos (A + B) = 1/2`, find A and B.

Answer 1

Step 1: Formulate the equations

From trigonometric standard values, we identify the angles that result in a value of `1/2` for both sine and cosine functions:

Since `sin 30^circ = 1/2`, the first equation is:

A – B = 30°   ...(i)

Since `cos 60^circ = 1/2`, the second equation is:

A + B = 60°   ...(ii)

Step 2: Solve the linear system

We can find the values of A and B by solving these two simultaneous equations:

1. Add equations (i) and (ii):

(A – B) + (A + B) = 30° + 60°

2A = 90°

A = 45°

2. Substitute A = 45° into equation (ii):

45° + B = 60°

B = 60° – 45°

B = 15°

Verification

sin (45° – 15°) = sin 30° 

= `1/2`

cos (45° + 15°) = cos 60° 

= `1/2`

The values are A = 45° and B = 15°.