simplify sinA+sin(A+B)+sin(A+2*B)+........+sin(A+(n-1)*B)

=  {[2sin(B/2)sinA + 2sind(B/2)sin(A+B) + 2sind(B/2)sin(A+2B) ….............+ 2sin(B/2)sin(A+(n-1)B)] }1/ 2sin(B/2)

{2sinXsinY = cos(X-Y) – cos(X+Y}

​=  {[cos(A - (B/2)) – cos(A + (B/2)) + cos (A + (B/2)) – cos (A + (3B/2)) + cos (A + (3B/2) …................... cos(A + (n-3/2)B) – cos (A + (n - 1)/2B)] 1}/2sin (B/2)

= { [cos (A – B/2) – cos(A + (n-1)2B]}1/2sin(B/2)

= 1/sin(B/2) 2 sin((2A+(n-1)B)/2)sin(nB/2)

= {sin(nB/2)(sin( (1^st angle) + sin(last angle) )/2) }1/sin(B/2)

 

where n = number of terms