How would you go about solving something like sin(5pi/3) without a calculator? Would you use a unit circle?

Answer 1

yes
or knowing that 5π/3 radians is 300°
the angle in standard position is 60°, that is, if I construct the triangle with
60° in the first quadrant.
You should know from your unit circle that sin60° = √3/2
But 300° is in quadrant III, and in that quad the sine is negative, thus....
sin 5π/3 = -√3/2

Btw, knowing the ratios of sides in a standard 30-60-90 and a 45-45-90 degree triangle means you don't even need the unit circle.