# A triangle has sides A, B, and C. Sides A and B are of lengths 4 and 7, respectively, and the angle between A and B is (3pi)/8 . What is the length of side C?

Jun 24, 2018

$c = 6.6$ units

#### Explanation:

$\hat{C} = \frac{3 \pi}{8} , a = 4 , b = 7$

Law of cosines ${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$

$\therefore {c}^{2} = {4}^{2} + {7}^{2} - \left(2 \cdot 4 \cdot 7 \cdot \cos \left(\frac{3 \pi}{8}\right)\right)$

${c}^{2} = 43.57$

$c = 6.6$ units

Jun 24, 2018

$c = \sqrt{65 - 28 \cdot \sqrt{2 - \sqrt{2}}}$

#### Explanation:

Using the Theorem of cosines

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\gamma\right)$
and note that

$\cos \left(\frac{\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}$
then we get

${c}^{2} = 49 + 16 - 2 \cdot 7 \cdot 4 \cdot \cos \left(3 \cdot \frac{\pi}{8}\right)$
so we get

${c}^{2} = 65 - 28 \sqrt{2 - \sqrt{2}}$