# A triangle has sides A,B, and C. If the angle between sides A and B is pi/4, the angle between sides B and C is pi/12, and the length of B is 7, what is the area of the triangle?

$5.177459202 \setminus \setminus {\textrm{u n i t}}^{2}$

#### Explanation:

The third angle between sides A & C in given triangle is given as

$= \setminus \pi - \setminus \frac{\pi}{4} - \setminus \frac{\pi}{12} = \frac{2 \setminus \pi}{3}$

Applying Sine rule in given triangle as follows

$\setminus \frac{B}{\setminus \sin \left(\frac{2 \setminus \pi}{3}\right)} = \setminus \frac{C}{\setminus \sin \left(\setminus \frac{\pi}{4}\right)}$

$\setminus \frac{7}{\setminus \frac{\sqrt{3}}{2}} = \setminus \frac{C}{\frac{1}{\setminus} \sqrt{2}}$

$C = 7 \setminus \sqrt{\frac{2}{3}}$

Now, the area of given triangle with two sides $B = 7$ & $C = 7 \setminus \sqrt{\frac{2}{3}}$ including an angle $\setminus \frac{\pi}{12}$ is

$= \frac{1}{2} B C \setminus \sin \left(\setminus \frac{\pi}{12}\right)$

$= \frac{1}{2} \left(7\right) \left(7 \setminus \sqrt{\frac{2}{3}}\right) \setminus \frac{\setminus \sqrt{3} - 1}{2 \setminus \sqrt{2}}$

$= \frac{49}{4} \left(1 - \frac{1}{\setminus} \sqrt{3}\right)$

$= 5.177459202 \setminus \setminus {\textrm{u n i t}}^{2}$