# A triangle has sides A, B, and C. Sides A and B are of lengths 2 and 5, respectively, and the angle between A and B is pi/4. What is the length of side C?

Sep 27, 2016

$\therefore C = \approx 3.855 .$

#### Explanation:

We use the Cosine-Rule for $\Delta A B C$, which states,

${C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos \left(\angle \left(A , B\right)\right) ,$

where, by "$\angle \left(A , B\right)$, we mean the angle btwn. sides A & B..

$\therefore {C}^{2} = {2}^{2} + {5}^{2} - 2 \cdot 2 \cdot 5 \cos \left(\frac{\pi}{4}\right)$

$= 29 - 20 \left(\frac{1}{\sqrt{2}}\right) = 29 - 10 \sqrt{2}$

$\approx 29 - 10 \left(1.414\right) = 29 - 14.14 = 14.86$

$\therefore C \approx 3.855$