Final answer:
The values of the trigonometric functions when csc t = 3 and the terminal point of t is in quadrant 2 are sin t = 1/3, cos t = -√(2/3), and tan t should be -√(2)/4.
Step-by-step explanation:
You want to find the values of trigonometric functions given that cosecant (csc t) equals 3 and the terminal point of t is in quadrant 2. In this quadrant, sine (sin t) is positive, and cosine (cos t) and tangent (tan t) are negative. Since csc t is the reciprocal of sin t, it means that sin t = 1/3, which is positive as expected for quadrant 2. Using the Pythagorean identity sin² t + cos² t = 1, we can find that cos t = -√(1 - (1/3)²) = -√(8/9) = -√(2/3).
Since tan t = sin t/cos t, we can calculate tan t as (-1/3)/(-√(2/3)) = 1/√(8) = √(2)/4. The negative signs cancel out since both sine and cosine are negative in the second quadrant, making the tangent positive as it should be. But since we are in the second quadrant, tangent should be negative, so there is a mistake here. Correcting for this, the tan t in quadrant 2 should be negative: tan t = -√(2)/4.