Q. 143.9( 9 Votes )

If A = 60° and B = 30°, verify that:

(i) sin (A + B) = sin A cos B + cos A sin B

(ii) cos (A + B) = cos A cos B — sin A sin B

Answer :

(i) To verify: sin (A + B) = sin A cos B + cos A sin B


If A = 60° and B = 30°, then


To verify: sin 90° = sin 60° cos 30° + cos 60° sin 30°


Consider R.H.S. = sin 60° cos 30° + cos 60° sin 30°


= (√3/2) × (√3/2) + (1/2)(1/2)


= (3/4) + (1/4)


= 4/4


=1


Consider L.H.S. = sin 90° = 1


L.H.S. = R.H.S.


Hence, verified.


(ii) To verify: cos (A + B) = cos A cos B — sin A sin B


If A = 60° and B = 30°, then


To verify: cos (90°) = cos 60° cos 30° — sin60° sin 30°


Consider R.H.S. = cos 60° cos 30° - sin 60° sin 30°


= (1/2) × (√3/2) - (√3/2)(1/2)


= (√3/4) - (√3/4)


= 0 = cos 90° = L.H.S.


L.H.S. = R.H.S.


Hence, verified.


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