Question

If sin (A + B) = sin A cos B + cos A sin B and cos (A – B) = cos A cos B + sin A sin B, find the values of (i) sin 75° (ii) cos 15°.

HINT: Take A = 45° and B = 30°.

Answer 1

Taking A = 45° and B = 30°, we have

(i) sin 75° = sin (45° + 30°)

= sin 45° cos 30° + cos 45° sin 30°

= `(1/sqrt(2) xx sqrt(3)/2 + 1/sqrt(2) xx 1/2)`

= `(sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2)))`

= `((sqrt(3) + 1)/(2sqrt(2)))`

(ii) cos 15° = cos (45° – 30°)

= cos 45° cos 30° + sin 45° sin 30°

= `(1/sqrt(2) xx sqrt(3)/2 + 1/sqrt(2) xx 1/2)`

= `(sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2)))`

= `((sqrt(3) + 1)/(2sqrt(2)))`