Question

A triangle has sides A, B, and C. Sides A and B are of lengths `6` and `3`, respectively, and the angle between A and B is `pi/4`. What is the length of side C?

Answer 1

`C = 3sqrt(5-2sqrt2)` or `~~4.42`

Explanation:

When you have the lengths of two sides of a triangle and the angle between, then you can solve the missing side with the law of cosines.

We want side `C`, which can be solved by the Law of Cosines formula `C = sqrt(A^2 + B^2 - 2(A)(B)cosc`, where `cosc` is the angle opposite to side `C`.

Let's substitute in the values and solve:
`C = sqrt((6)^2 + (3)^2 - 2(6)(3)cos(pi/4)`

Now simplify:
`C = sqrt(36 + 9 - 36(sqrt2/2)`

`C = sqrt(45 - 18sqrt2)`

`C = sqrt(9(5-2sqrt2)`

`C = sqrt9 * sqrt(5-2sqrt2)`

`C = 3sqrt(5-2sqrt2)`

You can leave it like that, but if you want the answer to be in decimal form, it is `~~4.42` (rounded to the nearest hundredth's place).

Hope this helps!