# A triangle has sides A, B, and C. Sides A and B are of lengths 2 and 8, respectively, and the angle between A and B is pi/3. What is the length of side C?

Jan 10, 2016

$C = 2 \sqrt{13}$

#### Explanation:

color(blue)("Test if "/_b = pi/2 -> 90^0

Known that $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$

If and only if ( iff ) $\angle b = \frac{\pi}{2}$ then $\frac{A}{B} \to \frac{1}{2}$
But $\frac{A}{B} = \frac{2}{8} = \frac{1}{4} \implies \angle b \ne \frac{\pi}{2}$

color(blue)("Shown that "/_b !=pi/2
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Solve using the Cosine rule}}$

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

$\implies {C}^{2} = {2}^{2} + {8}^{2} - 2 \left(2\right) \left(8\right) \cos \left(\frac{\pi}{3}\right)$

${C}^{2} = 4 + 64 - 32 \left(\frac{1}{2}\right)$

${C}^{2} = 52$

$C = 2 \sqrt{13}$