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#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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