Question

Find the exact value of
sin(cos^-1(-1/3))

Answer 1

(cos^-1(-1/3)) is essentially the same as saying cos(θ)=-1/3 (-1 < x < 0)

So, theta falls in the second quadrant and sin(θ)>0 (i.e. positive)

here's an acronym to you determine when sin, cos, and tan are positive:
"All S-tudents T-ake C-alculus"
All=first quadrant all positive
S-tudents=second quadrant sin is positive
T-ake=third quadrant tan is positive
C-alculus=fourth quadrant cos is positive

sin^2(θ) + cos^2(θ) = 1
sin^2(θ) + (-1/3)^2 = 1
then, solve for sin(θ)