How do you solve sin (5pi/3) without a calculator?

Well, you're just suppose to know where 5pi/3 is on the unit circle.

That is in the fourth quadrant with coordinates (1/2, - root3/2)

So sin of 5pi/3 would be = to - root3/2

very simple, we use the formula sin (a+b) = sinacosb + sinbcosa and sin (20) = 2sinacosa

5pi = 2pi/3+3pi/3,

First, we use sin (a+b) = sinacosb + sinbcosa

sin (5pi/3) = sin (2pi/3+3pi/3) = sin (2pi/3+pi) = sin (2pi/3) cos (pi) + sin (pi) cos (2pi/3)

but we know that sin (pi) = 0, and cos (pi) = - 1, so sin (5pi/3) = - sin (2pi/3)

now, use sin (2a) = 2sinacosa, sin (5pi/3) = - sin (2pi/3) = - 2sin (pi/3) cos (pi/3)

sin (5pi/3) = - 2sin (pi/3) cos (pi/3)

sin (pi/3) = 0.86, cos (pi/3) = 0.5, finally we have sin (5pi/3) = - 0.86 x 0.5 = - 0.43