# A triangle has sides A, B, and C. Sides A and B are of lengths 6 and 8, respectively, and the angle between A and B is pi/4. What is the length of side C?

Dec 22, 2015

$c = 5.67$ rounded to hundredth.

#### Explanation:

Cosine Law

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\theta\right)$

In our problem $a = 6$ and $b = 8$ and we are also given the angle between as $\frac{\pi}{4}$

To find $c$ we plug in the values in the cosine law and we get.

${c}^{2} = {6}^{2} + {8}^{2} - 2 \cdot 6 \cdot 8 \cdot \cos \left(\frac{\pi}{4}\right)$
${c}^{2} = 36 + 64 - 96 \left(\frac{\sqrt{2}}{2}\right)$
${c}^{2} = 100 - 48 \cdot \sqrt{2}$
${c}^{2} = 32.117749006091437657518941237934$ using calculator
$c = \sqrt{32.117749006091437657518941237934}$
$c = 5.6672523330174242081448041317437$

$c = 5.67$ rounded to hundredth.