# A triangle has sides A, B, and C. The angle between sides A and B is (3pi)/4. If side C has a length of 24  and the angle between sides B and C is pi/12, what are the lengths of sides A and B?

Feb 18, 2018

Lengths of sides color(red)(a = 8.7846, b = 16.9706

#### Explanation:

Given : $\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{3 \pi}{4} , c = 24$

To find $a , b$

Third angle $\hat{B} = \pi - \frac{3 \pi}{4} - \frac{\pi}{12} = \frac{\pi}{6}$

$\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C$

$a = \frac{c \cdot \sin A}{\sin} C = \frac{24 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{3 \pi}{4}\right) = \textcolor{p u r p \le}{8.7846}$

b = (c * sin B) / sin C = (24 * sin (pi/6)) / sin ((3pi)/4) = color (purple)(16.9706