Question

A triangle has sides A, B, and C. Sides A and B are of lengths `4` and `2`, respectively, and the angle between A and B is `(3pi)/4`. What is the length of side C?

Answer 1

`abs(C)=color(green)(2sqrt(5+2sqrt(2)))`

Explanation:

If `c` is the angle opposite side `C`
then `c=(3pi)/4` (given)

and by the Law of Cosines
`color(white)("XXX")C^2=A^2+B^2-2ABcos(c)`

In this case
`color(white)("XXX")C^2=4^2+2^2-2 * 4 * 2 * (-sqrt(2)/2)`

`color(white)("XXXX")=20+8sqrt(2)`

`color(white)("XXXX")=2^2(5+2sqrt(2))`

`rArr`
`color(white)("XXX")abs(C)=2sqrt(5+2sqrt(2))`

(or using a calculator to evaluate an approximation)
`color(white)("XXX")abs(C)~~5.595865`