# A triangle has sides A, B, and C. The angle between sides A and B is pi/3 and the angle between sides B and C is pi/12. If side B has a length of 14, what is the area of the triangle?

Area$= 22.740979141751 \text{ }$square units

#### Explanation:

From the given:
Angles $A = \frac{\pi}{12} = {15}^{\circ}$ and $C = \frac{\pi}{3} = {60}^{\circ}$
Therefore $B = {105}^{\circ}$
but side $b = 14$

We only need one side to solve for the Area.

By the sine law
$\frac{a}{\sin} A = \frac{b}{\sin} B$

$a = \frac{b \cdot \sin A}{\sin} B$

$a = \frac{14 \cdot \sin {15}^{\circ}}{\sin} {105}^{\circ}$

$a = 3.7512886940358$

Now we can use the formula for area:

Area$= \frac{1}{2} \cdot a \cdot b \cdot \sin C$
Area$= \frac{1}{2} \cdot \left(3.7512886940358\right) \cdot 14 \cdot \sin {60}^{\circ}$

Area$= 22.740979141751 \text{ }$square units

God bless....I hope the explanation is useful.