A triangle has sides A,B, and C. If the angle between sides A and B is #(5pi)/8#, the angle between sides B and C is #pi/12#, and the length of B is 3, what is the area of the triangle?

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Answer 1 · 2016-02-06T16:01:19

Area #= 1.356" units" ^2# to 3 decimal places

Tony B

#color(blue)("Method")#

Find h using the sin rule. Then use h to determine the area.

#color(blue)("Solution")#

Target is to be able to apply #C/(sin(c))=B/(sin(b))#

To do this we need to find #/_cba#

The sum of the internal angles in a triangle is #180^o#

#=> /_cba=pi-(5pi)/8-(pi)/12=(7pi)/24" " (52.5 ^o)#

Thus we have

#C/(sin((5pi)/8)) = 3/(sin((7pi)/24))#

#C= 3 xxsin((5pi)/8)/(sin((7pi)/24))#

#h= C sin(pi/12)#

#h= 3 xxsin((5pi)/8)/(sin((7pi)/24))xxsin(pi/12)#

#Area = B/2xxh" "=" "9/2 xxsin((5pi)/8)/(sin((7pi)/24))xxsin(pi/12)#

#9/2xx(sin(112.5^o))/(sin(52.5^o))xxsin(15^o)#

#=1.356 " units"^2#