# A triangle has sides A, B, and C. Sides A and B are of lengths 9 and 3, respectively, and the angle between A and B is pi/6. What is the length of side C?

Jan 29, 2016

$C \approx 6.575$

#### Explanation:

Let the perpendicular height of the triangle be $h$. This intersects side $A$ so that lengths $x$ and $y$ add up to $A$

$A = 9$
$B = 3$

$\sin \left(\frac{\pi}{6}\right) = \frac{h}{3}$
$\therefore h = 3 \sin \left(\frac{\pi}{6}\right) = 3 \cdot 0.5 = 1.5$

$\cos \left(\frac{\pi}{6}\right) = \frac{x}{3}$
$\therefore x = 3 \cos \left(\frac{\pi}{6}\right)$

$y = 9 - 3 \cos \left(\frac{\pi}{6}\right)$

Let the angle between $A$ and $C$ be $\theta$. Then
$\tan \theta = \frac{h}{y} = \frac{1.5}{9 - 3 \cos \left(\frac{\pi}{6}\right)}$

$\theta = {\tan}^{-} 1 \left(\frac{1.5}{9 - 3 \cos \left(\frac{\pi}{6}\right)}\right)$

$\frac{h}{C} = \sin \theta$
$\therefore C = \frac{1.5}{\sin} \theta$

From here it is just a lot of grunt work with a calculator to get the length.

$C \approx 6.575$