Find the exact value of the trigonometric function given that sin u = -5/13 and cos v = -20/29. (Both u and v are in Quadrant III.) tan(u − v)

Answer 1

Use the angle difference identity for tangent:

(of course substituting u and v for x and y)

To use this, you needs to determine tan(u) and tan(v).

Since sin(u) = opp/hyp = -5/13, we can think of the triangle with the opposite side = -5 and the hypotenuse = 13 (ignoring the positive and negative signs for the moment gives opp = 5). Use the Pythagorean Theorem to solve for the adjacent side. You will get adj = 12. Since we are in Q III, this would need to be a -12. Since tan(u) = opp/adj then tan(u) = -5/-12 = 5/12

Since cos(v) = adj/hyp = -20/29, we can think of the triangle with the adjacent side = -20 and the hypotenuse = 29 (ignoring the positive and negative signs for the moment gives adj = 20). Use the Pythagorean Theorem to solve for the opposite side. You will get opp = 21. Since we are in Q III, this would need to be a -21. Since tan(v) = opp/adj then tan(v) = -21/-20 = 21/20

Plugging these in the tangent angle difference identity:

tan(u - v) = (5/12 - 21/20)/(1 + 5/12•21/20)

tan(u - v) = (-19/30)/(1 + 7/16)

tan(u - v) = (-19/30)/(23/16)

tan(u - v) = (-152/345)