Question

A triangle has sides A, B, and C. The angle between sides A and B is `(3pi)/4`. If side C has a length of `8` and the angle between sides B and C is `pi/12`, what are the lengths of sides A and B?

Answer 1

`A = 2.93`
`B = 5.66`

Explanation:

Given two angles, the third one in a triangle is fixed. In this case it is `2pi/12`. The shortest side length will be opposite the smallest angle, which is `pi/12` in this case. We know that the side of length 8 is opposite the `9pi/12` corner.

We now have three angles and a side, and can calculate the other sides using the Law of Sines, and then calculate the height for the area.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-sines

https://www.mathsisfun.com/algebra/trig-solving-asa-triangles.html

`a/(sin(pi/12)) = c/sin C = 8/(sin(9pi/12))`
`b/(sin(2pi/12)) = c/sin C = 8/(sin(9pi/12))`

`a xx (sin(9pi/12)) = 8 xx (sin(pi/12))`

`b xx (sin(9pi/12)) = 8 xx (sin(2pi/12))`

`a xx 0.707 = 8 xx 0.259` ; `a = 2.93`
`b xx 0.707 = 8 xx 0.50` ; `b = 5.66`