Finding exact values of sec and cot

Question

Let theta be an angle in quadrant 3 such that sin theta equals -3/5. Find the exact values of sec theta and cot theta.

Answer 1

sin(theta) = y/r = -3/5 ---> y = -3; r = 5; with x^2 + y^2 = r^2, we have x^2 = r^2 - y^2 = 5^2 - 3^2 = 25 - 9 = 16, so that x = +/- 4; since x < 0 in Q3, we have x = -4; then: cos(theta) = x/r = -4/5; so that sec(theta) = 1/cos(theta) = -5/4; and cot(theta) = cos(theta)/sin(theta) = (-4/5)/(-3/5) = 4/3