"sin^-1(sin330)" I know the answer is π/6, or 30° but how do you figure this out without a calculator? studying for my final thank you!?

sin^-1 (sin 330) = -30

sin^-1 (sin x) = x if -90<x<90 (or -pi/2 and pi/2)

if x is outside this range...
work with these rules...
sin x = sin (180 - x)
sin x = sin (360 + x)
-sin x = sin -x

or....
sin 330 = -1/2
sin^-1 (-1/2) = -30

inverse sine undoes the sin function so it leaves you with the angle of 330 so the angle is not pi/6 its -pi/6 .
330 is in the fourth quadrant, the reference angle is 30 from the terminal side CLOCKWISE meaning the angle in negative 30 measured from standard position

if the sin x = 1 you find x by taking the inverse sin , because it UNDOES the function. sin^-1 is the inverse function of sin. so just like every other inverse. f^-1 ( f(x) ) = x

in fact it is not what you you think.....- π / 6 or - 30°......sin 330° < 0...sin 330° = sin ( - 30° )