# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/12, the angle between sides B and C is (5pi)/12, and the length of B is 10, what is the area of the triangle?

Aug 17, 2017

The area of the triangle is $12.50$ sq.unit .

#### Explanation:

The angle between sides $A \mathmr{and} B$ is $\angle c = \frac{\pi}{12} = \frac{180}{12} = {15}^{0}$

The angle between sides $B \mathmr{and} C$ is  /_a =(5 pi)/12=(5*180)/12= 75^0

The angle between sides $C \mathmr{and} A$ is $\angle b = 180 - \left(75 + 15\right) = {90}^{0}$

We can find side $A$ by aplying sine law  A/sina = B/sinb ; B=10

$A = B \cdot \sin \frac{a}{\sin} b = 10 \cdot \sin \frac{75}{\sin} 90 \approx 9.66$

Now we have Sides $A \left(9.66\right) , B \left(10\right)$ and their included angle

$\angle c = {15}^{0}$. The area of the triangle is ${A}_{t} = \frac{A \cdot B \cdot \sin c}{2}$

or ${A}_{t} = \frac{10 \cdot 9.66 \cdot \sin 15}{2} = 12.50$ sq.unit [Ans]