# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/2. If side C has a length of 28  and the angle between sides B and C is pi/12, what is the length of side A?

Aug 11, 2018

$A = 7 \left(\sqrt{6} - \sqrt{2}\right) \approx 7 \left(1.0353\right) = 7.2471$.

#### Explanation:

Let us denote by $\angle \left(B , C\right)$ the angle between sides $B \mathmr{and} C$.

Using the Sine-Rule, then, we have,

$\frac{A}{\sin \angle \left(B , C\right)} = \frac{B}{\sin \left(\angle C , A\right)} = \frac{C}{\sin \left(\angle A , B\right)}$.

$\therefore \frac{A}{\sin \angle \left(B , C\right)} = \frac{C}{\sin \left(\angle A , B\right)}$.

$\therefore \frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{28}{\sin} \left(\frac{\pi}{2}\right) = \frac{28}{1}$.

$\therefore A = 28 \sin \left(\frac{\pi}{12}\right) = 28 \left\{\frac{\sqrt{3} - 1}{2 \sqrt{2}}\right\}$.

$\Rightarrow A = 7 \left(\sqrt{6} - \sqrt{2}\right) \approx 7 \left(1.0353\right) = 7.2471$.