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?

I diagram usually helps to understand what is happening.

`color(blue)("Plan")`

Find `/_AB` using sum of internal angles.
Use sin rule to determine length of side C
Determine length of h using sin
Area = `C/2xxh`
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`color(blue)("To determine "/_AB)`

Sum of internal angles of a triangle is `180^o ->pi`

`color(brown)(=>/_AB = pi-(7pi)/24 -(5pi)/8 = pi/12 -> 15^o)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine length of h")`

`color(green)("Value of h for condition 1")`

`Axxsin( (7pi)/24)=h`

`color(blue)(h=8xx sin((7pi)/24)color(red)(~~6.3568...))`

`color(green)("Value of h for condition 2")`

`Bxxsin(pi-(5pi)/8)=h`

`color(blue)(h= 3xxsin(pi-(5pi)/8)color(red)(~~ 2.7716...))`

`color(red)("Contradiction!")`

`underline(color(red)("We have two different values for h"))`

`color(blue)("Assumption: Length of A is the correct length")`

`color(blue)(=>h~~6.3568...)`

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

`color(blue)("Determine the length of C")`

`A/sin((5pi)/8) =C/sin(Pi/12)`

`=>C=(Axxsin(pi/12))/(sin((5pi)/8)) =(8xxsin(pi/12))/(sin((5pi)/8))`

`color(blue)(C~~2.2411...)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine area")`

Area = `C/2xxh " "->" Area"~~2.2411/2xx6.3568`

`color(blue)(=> Area ~~7.1231" units"^2)` to 4 decimal places