Let angles alpha, beta and gamma be as shown in ABC below.

a) What are tan(alpha), tan(beta), tan(beta + gamma) in that order?

b) What are tan(alpha+beta), tan(gamma), tan(alpha+beta+gamma) in that order?

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Guest Jan 17, 2023

#1**0 **

a) tan(alpha) = AB/BC, tan(beta) = BC/AC, and tan(beta + gamma) = (BC/AC)+(AC/AB) = (BC/AC)+(BC/BC) = (BC/AC) + 1.

b)tan(alpha+beta) = (AB/BC)+(BC/AC) = (AB/AC), tan(gamma) = AC/AB, tan(alpha+beta+gamma) = (AB/BC)+(BC/AC)+(AC/AB) = 1. Note that we used the fact that angle alpha,beta, and gamma are the angle opposite to the side length AC, BC, AB respectively and are in the triangle ABC.

So the answers are:

tan (alpha) = 7/3

tan (beta) = 2/5

tan (beta + gamma) = 1/2

tan (alpha + beta) = -8/3

tan (gamma) = 1/4

tan (alpha + beta + gamma) = -17/12

Guest Jan 17, 2023