Given cot u=2/5, 0<u<pi/2 and cos v = -3/5, pi<v<3pi/2, find tan(u+v)

Question

When there is a given, how do you plug that in to find an answer?

Answer 1

We will use the trigonometric identify tan(u+v) = tan u + tan v / (1 - tan u * tan v) to find the answer.

This means we need to find tan u and tan v.

tan u is the inverse of cot u, so since cot u = 2/5, tan u = 5/2 = 2.5. Since we are in the first quadrant, tan u will be positive.

Basic trig identity: cos v ** 2 + sin v **2 = 1

Since cos v = -3/5, then per the above identity, (-3/5)**2 + sin v**2 = 1

9/25 + sin v**2 = 1

sin v**2 = 16/25

sin v = +/- 4/5

Since pi < v < 3/2 pi, we are in the 3rd quadrant, where the sin is negative, so sin v = -4/5.

tan v = sin v/ cos v = -4/5 / -3/5 = 4/3 = 1.33

plugging the computed values for tan u and tan v back into the trigonometric identity:

tan(u+v) = tan u + tan v / (1 - tan u * tan v)

= (2.5 + 1.3) / (1 - 2.5 * 1.3) = -1.68