Final answer:
To find sin(s - t), we use the identity sin(a - b) = sin a cos b - cos a sin b. We calculate sin s and cos t using the Pythagorean identity, then apply the formula to get sin(s-t) = 2√2/3 + 1/6.
Step-by-step explanation:
To find sin(s - t) given that cos s = 1/3, with s in quadrant I, and sin t = -1/2, with t in quadrant IV, we use the fact that sin(a - b) = sin a cos b - cos a sin b. Since s is in quadrant I, both sin s and cos s are positive. Since t is in quadrant IV, cos t is positive and sin t is negative.
Given that cos s = 1/3, we can find sin s using the Pythagorean identity sin² s + cos² s = 1. This gives us sin s = √(1 - (1/3)²) = √(1 - 1/9) = √(8/9) = √8/3.
Given sin t = -1/2 and t is in quadrant IV, we again use the Pythagorean identity to find cos t. Sin² t + cos² t = 1, thus cos t = √(1 - sin² t) = √(1 - (1/2)²) = √(1 - 1/4) = √(3/4) = √3/2.
Now we can find sin(s-t) using the formula: sin(s-t) = sin s cos t - cos s sin t = (√8/3)(√3/2) - (1/3)(-1/2) = 2√2/3 + 1/6.