# How do you find the exact value of the third side given triangle DEF, d=sqrt3, e=5, and mangleF=pi/6?

Mar 25, 2018

color(indigo)("Exact value of length of side " f = sqrt13

#### Explanation:

Given : $d = \sqrt{3} , e = 5 , \hat{F} = \frac{\pi}{6} , \text{ To find length of side f}$

$\text{Applying Law of cosines}$,

$f = \sqrt{{d}^{2} + {e}^{2} - \left(2 \cdot d \cdot e \cdot \cos F\right)}$

f = sqrt((sqrt3)^2 + 5^2 - (2 * sqrt3 * 5 * cos (pi/6)

$f = \sqrt{3 + 25 - \left(2 \cdot \sqrt{3} \cdot 5 \cdot \left(\frac{\sqrt{3}}{2}\right)\right)} = \sqrt{28 - 15} = \sqrt{13}$