# 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 12  and the angle between sides B and C is pi/12, what is the length of side A?

The three sides are:

$A = 3 \sqrt{2} \left(\sqrt{3} + 1\right)$

$B = 3 \sqrt{2} \left(\sqrt{3} - 1\right)$

$C = 12$

#### Explanation:

$C = 12$

${\theta}_{C} = \frac{\pi}{2}$

${\theta}_{A} = \frac{\pi}{12}$

The triangle is a right angled triangle with C as hypotenuse,

Adjacent side $A = C \cos {\theta}_{A}$

$= 12 \cos \left(\frac{\pi}{12}\right)$

$\cos \left(\frac{\pi}{12}\right) = \frac{\sqrt{3} + 1}{2 \sqrt{2}}$

$A = 12 \times \frac{\sqrt{3} + 1}{2 \sqrt{2}} = 3 \sqrt{2} \left(\sqrt{3} + 1\right)$

$B = C \sin \left(\frac{\pi}{12}\right)$

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

$A = 12 \times \frac{\sqrt{3} - 1}{2 \sqrt{2}} = 3 \sqrt{2} \left(\sqrt{3} - 1\right)$

The three sides are:

$A = 3 \sqrt{2} \left(\sqrt{3} + 1\right)$

$B = 3 \sqrt{2} \left(\sqrt{3} - 1\right)$

$C = 12$