# A triangle has sides A, B, and C. Sides A and B have lengths of 8 and 3, respectively. The angle between A and C is (7pi)/24 and the angle between B and C is  (5pi)/8. What is the area of the triangle?

Mar 21, 2016

The question has a wrong value! Assuming A to be the correct length then
$\textcolor{b l u e}{\implies A r e a \approx 7.1231 {\text{ units}}^{2}}$ to 4 decimal places
$\textcolor{w h i t e}{.}$
$\textcolor{red}{\text{At least you will see the method}}$

#### Explanation:

I diagram usually helps to understand what is happening.

$\textcolor{b l u e}{\text{Plan}}$

Find $\angle A B$ using sum of internal angles.
Use sin rule to determine length of side C
Determine length of h using sin
Area = $\frac{C}{2} \times h$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{To determine } \angle A B}$

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

$\textcolor{b r o w n}{\implies \angle A B = \pi - \frac{7 \pi}{24} - \frac{5 \pi}{8} = \frac{\pi}{12} \to {15}^{o}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine length of h}}$

$\textcolor{g r e e n}{\text{Value of h for condition 1}}$

$A \times \sin \left(\frac{7 \pi}{24}\right) = h$

$\textcolor{b l u e}{h = 8 \times \sin \left(\frac{7 \pi}{24}\right) \textcolor{red}{\approx 6.3568 \ldots}}$

$\textcolor{g r e e n}{\text{Value of h for condition 2}}$

$B \times \sin \left(\pi - \frac{5 \pi}{8}\right) = h$

$\textcolor{b l u e}{h = 3 \times \sin \left(\pi - \frac{5 \pi}{8}\right) \textcolor{red}{\approx 2.7716 \ldots}}$

$\textcolor{red}{\text{Contradiction!}}$

$\underline{\textcolor{red}{\text{We have two different values for h}}}$

$\textcolor{b l u e}{\text{Assumption: Length of A is the correct length}}$

$\textcolor{b l u e}{\implies h \approx 6.3568 \ldots}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

$\frac{A}{\sin} \left(\frac{5 \pi}{8}\right) = \frac{C}{\sin} \left(\frac{\Pi}{12}\right)$

$\implies C = \frac{A \times \sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{5 \pi}{8}\right)} = \frac{8 \times \sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{5 \pi}{8}\right)}$

$\textcolor{b l u e}{C \approx 2.2411 \ldots}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine area}}$

Area = $\frac{C}{2} \times h \text{ "->" Area} \approx \frac{2.2411}{2} \times 6.3568$

$\textcolor{b l u e}{\implies A r e a \approx 7.1231 {\text{ units}}^{2}}$ to 4 decimal places