sin240+sin120+sin150+sin210=?

To determine the value of the sum, we'll apply the formula
that transforms the sum of sine functions into a
product:

sin a + sin b = 2
sin[(a+b)/2]*cos[(a-b)/2]

We'll combine the first and
the second term:

sin 120 + sin 240 = 2
sin[(120+240)/2]*cos[(120-240)/2]

sin 120 + sin 240 = 2
sin[(360)/2]*cos[(-120)/2]

sin 120 + sin 240= 2 sin 180*cos
(-60)

But sin 180 = 0,
so:

sin 120 + sin 240 =
0

We'll combine the third and the last
term:

sin150+sin210 = 2
sin[(150+210)/2]*cos[(150-210)/2]

sin150+sin210 = 2 sin
(360/2) *cos (-60/2)

sin150+sin210 = 2 sin 180*cos
(-30)

But cos(-30) = cos 30,because the function cosine is
an even function.

sin150+sin210 = 2*0*cos
30

sin150+sin210 = 0

So, the
value of the sum
is:

sin240+sin120+sin150+sin210 = 0+0 =
0