Question

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

Answer 1

`color(indigo)("Length of sides " A = B = 15.68`

Explanation:

Given two angles, the third one in a triangle is fixed. In this case it is `3pi/8` (and we have an isosceles triangle!). The shortest side length will be opposite the smallest angle, which is `2pi/8` in this case. We know that the side of length 12 is opposite the `2pi/8` 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((3pi)/8)) = c/sin C = 12/(sin((2pi)/8))`
`b/(sin((3pi)/8)) = c/sin C = 12/(sin((2pi)/8))`

`a xx sin((2pi)/8) = 12 xx sin((3pi)/8)`

`b xx sin((2pi)/8) = 12 xx sin((3pi)/8)`

Since `hat A = hat B, " it's an isosceles triangle"`

`a xx 0.707 = 12 xx 0.924` ; `color(brown)(a = 15.68 = b`