# A triangle has sides A, B, and C. Sides A and B are of lengths 7 and 5, respectively, and the angle between A and B is (pi)/8 . What is the length of side C?

Sep 29, 2016

≈3.054" units"

#### Explanation:

Given a triangle, where 2 sides and the angle between them are known, in this case A and B and we wish to calculate the length of the third side C, use the $\textcolor{b l u e}{\text{cosine rule}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{C}^{2} = {A}^{2} + {B}^{2} - \left(2 A B \cos \left(\text{angle between them}\right)\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

here A = 7 , B = 5 and angle between them $= \frac{\pi}{8}$

substitute these values into the 'cosine rule'

${C}^{2} = {7}^{2} + {5}^{5} - \left(2 \times 7 \times 5 \times \cos \left(\frac{\pi}{8}\right)\right)$

$= 49 + 25 - \left(64.672 \ldots\right)$

=74-(64.672...)≈9.328

Now  C^2≈9.328rArrC=sqrt(9.328)≈3.054