# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 3, what is the area of the triangle?

##### 1 Answer
Jan 15, 2016

$\textcolor{g r e e n}{{\text{area " = 1.125 " units"^2 -> 1 1/8 " units}}^{2}}$

#### Explanation:

$\textcolor{b l u e}{\text{Assumption: }}$

As $\pi$ is used in the angular measure it is assumed that the unit is radians. (Not stated)

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$\textcolor{b l u e}{\text{Method Plane}}$

Determine $\angle c b a$

Using Sine Rule and $\angle c b a$ determine length of side A
Determine h using $h = A \sin \left(\frac{5 \pi}{12}\right)$
Determine area $h \times \frac{B}{2}$

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$\textcolor{b l u e}{\text{To determine} \angle c b a}$

Sum internal angles of a triangle is ${180}^{0} = \pi \text{ radians}$

$\implies \angle c b a = \pi - \frac{5 \pi}{12} - \frac{\pi}{12}$
$\textcolor{b r o w n}{\angle c b a = \frac{\pi}{2} \to {90}^{o}}$

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$\textcolor{b l u e}{\text{Determine length of A}}$

Using $\frac{B}{\sin \left(b\right)} = \frac{A}{\sin} \left(a\right)$

$\implies \frac{3}{\sin \left(\frac{\pi}{2}\right)} = \frac{A}{\sin} \left(\frac{\pi}{12}\right)$

$\implies A = \frac{3 \times \sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{\pi}{2}\right)}$

But $\sin \left(\frac{\pi}{2}\right) = 1$

$\textcolor{b l u e}{\implies A = 3 \times \sin \left(\frac{\pi}{12}\right)}$

$\textcolor{b r o w n}{\text{This is an exact value so keep it in this form for now to reduce error}}$ $\textcolor{b r o w n}{\text{on final calculation.}}$
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$\textcolor{b l u e}{\text{To determine h}}$

$h = A \sin \left(\frac{5 \pi}{12}\right)$

$\implies h = 3 \times \sin \left(\frac{\pi}{12}\right) \times \sin \left(\frac{5 \pi}{12}\right)$

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$\textcolor{b l u e}{\text{To determine area}}$

$\text{area } = \frac{B}{2} \times h$

$\text{area } = \frac{3}{2} \times 3 \times \sin \left(\frac{\pi}{12}\right) \times \sin \left(\frac{5 \pi}{12}\right)$

but $\sin \left(\frac{\pi}{12}\right) \times \sin \left(\frac{5 \pi}{12}\right) = \frac{1}{4}$

$\text{area } = \frac{3}{2} \times 3 \times \frac{1}{4}$

$\textcolor{g r e e n}{{\text{area " = 1.125 " units"^2 -> 1 1/8 " units}}^{2}}$