# A triangle has sides A, B, and C. Sides A and B are of lengths 7 and 4, respectively, and the angle between A and B is pi/6. What is the length of side C?

Jun 25, 2016

$C = \sqrt{65 - 28 \sqrt{3}} = \sqrt{16.51} \cong 4.06 .$
We use Cosine-rule for $\Delta L M N : {n}^{2} = {l}^{2} + {m}^{2} - 2 \cdot l \cdot m \cdot \cos \angle \left(l , m\right) .$
$\therefore {C}^{2} = {A}^{2} + {B}^{2} - 2 A \cdot B \cdot \cos \angle \left(A , B\right) = 49 + 16 - 56 \cos \left(\frac{\pi}{6}\right) = 65 - 56 \cdot \left(\frac{\sqrt{3}}{2}\right) = 65 - 28 \sqrt{3} \cong 65 - 48.49 = 16.51 .$
$\therefore C = \sqrt{16.51} \cong 4.06 .$