Find the exact value of sin(u+v) given that sin u=7/25 and cos v=-12/13

Question

Find the exact value of sin(u+v) given that sin u=7/25 and cos v=-12/13

(Both u and v are in Quadrant 2)

Answer 1

First, find the missing function values from a sketch and the Pythagorean Theorem: cos u = -24/25 and sin v = 5/13. Use the sum/difference identity sin(u+v) = sin u cos v + cos u sin v = (7/25)(-12/13) + (-24/25)(5/13) = -84/325 - 120/325 = - 204/325. Check your work: use inverse functions and the quadrant 2 fact to find the angles u and v, add them together, and calculate sin(u+v).