Question

Prove that 

(i) sin A + sin(120° + A) + sin (240° + A) = 0 

(ii) cos A + cos (120° +A) + cos (120° – A) = 0

Answer 1

(i) sin A + sin(120° + A) + sin (240° + A) 

= sin A+ sin 120° cos A + cos 120° sin A + sin 240° 

cos A + cos 240° sin A ..… (1)

Now,

By substituting these values in (1), we get,

(ii) cos 120° = cos (180° – 60°) = – cos 60° = -1/2 

LHS = cos A + cos (120° + A) + cos (120° – A) 

= cos A + cos 120° cos A – sin 120° sin A + cos 120° cos A + sin 120° sin A 

= cos A + 2 cos 120° cos A

cos A + 2(-1/2) cos A = cos A - cos A = 0 = RHS