# A triangle has sides A, B, and C. The angle between sides A and B is pi/6 and the angle between sides B and C is pi/12. If side B has a length of 1, what is the area of the triangle?

Aug 7, 2017

Area of the triangle is $0.09$ sq. unit.

#### Explanation:

The angle between sides $A \mathmr{and} B$ is $\angle c = \frac{\pi}{6} = \frac{180}{6} = {30}^{0}$.

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

The angle between sides $C \mathmr{and} A$ is $\angle b = 180 - \left(30 + 15\right) = {135}^{0}$

$B = 1$ . Applying sine law we get $\frac{A}{\sin} a = \frac{B}{\sin} b$

$\therefore A = B \cdot \left(\sin \frac{a}{\sin} b\right) = 1 \cdot \sin \frac{15}{\sin} 135 = 0.366$

Now we have $A = 0.366 , B = 1$ and their icluded angle

$c = {30}^{0}$:. Area of the triangle is ${A}_{t} = \frac{A \cdot B \cdot \sin c}{2}$ or

${A}_{t} = \frac{1 \cdot 0.366 \cdot \sin 30}{2} = \frac{0.366 \cdot \frac{1}{2}}{2} \approx 0.09$ sq. unit Ans]