# A triangle has sides A, B, and C. If the angle between sides A and B is (3pi)/8, the angle between sides B and C is (7pi)/12, and the length of B is 3, what is the area of the triangle?

Mar 25, 2016

$A r e a \approx 30.766 \text{ units"^2" }$to 3 dp

#### Explanation:

Sum of internal angles for a triangle is ${180}^{o} \to \pi \text{ radians}$

So $\angle A C = \frac{1}{24} \pi \to 7 \frac{1}{2} {\textcolor{w h i t e}{.}}^{o}$

$h = B \sin \left(\angle A B\right) = 3 \sin \left(\frac{3}{8} \pi\right) \approx 2.7716$

$\text{ } \frac{A}{\sin} \left(\angle B C\right) = \frac{B}{\sin} \left(\angle A C\right)$

$\implies A = \frac{B \sin \left(\angle B C\right)}{\sin} \left(\angle A C\right) \approx 22.201$

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$A r e a = \frac{A}{2} \times h$

$A r e a = \frac{B \sin \left(\angle B C\right)}{2 \sin \left(\angle A C\right)} \times B \sin \left(\angle A B\right)$

$A r e a = \frac{3 \sin \left(\angle B C\right)}{2 \sin \left(\angle A C\right)} \times 3 \sin \left(\angle A B\right) \approx 30.766 \text{ }$to 3 dp