Question

Find the exact value of the trigonometric expression when

Answer 1

Part of the value of sin(u) is cut off; I suspect it should be either sin(u) = -5/13 or sin(u) = -12/13, since (5, 12, 13) is a Pythagorean triple. I'll assume -5/13.

Expand the tan expression using the angle sum identities for sin and cos :

tan(u + v) = sin(u + v) / cos(u + v)

tan(u + v) = [sin(u) cos(v) + cos(u) sin(v)] / [cos(u) cos(v) - sin(u) sin(v)]

Since both u and v are in Quadrant III, we know that each of sin(u), cos(u), sin(v), and cos(v) are negative.

Recall that for all x,

cos²(x) + sin²(x) = 1

and it follows that

cos(u) = - √(1 - sin²(u)) = -12/13

sin(v) = - √(1 - cos²(v)) = -3/5

Then putting everything together, we have

tan(u + v)

= [(-5/13) • (-4/5) + (-12/13) • (-3/5)] / [(-12/13) • (-4/5) - (-5/13) • (-3/5)]

= 56/33

(or, if sin(u) = -12/13, then tan(u + v) = -63/16)