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?

#color(blue)("Assumption: ")#

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

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#color(blue)("Method Plane")#

Determine #/_cba#

Using Sine Rule and #/_cba# determine length of side A
Determine h using #h=Asin((5pi)/12)#
Determine area #hxxB/2#

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#color(blue)("To determine" /_cba)#

Sum internal angles of a triangle is #180^0 = pi" radians"#

#=>/_cba= pi-(5pi)/12-pi/12#
#color(brown)(/_cba = pi/2 -> 90^o)#

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#color(blue)("Determine length of A")#

Using #B/(sin(b))=A/sin(a)#

#=> 3/(sin(pi/2)) =A/sin(pi/12)#

#=> A= (3xxsin(pi/12))/(sin(pi/2))#

But #sin(pi/2) = 1#

#color(blue)(=> A = 3xxsin(pi/12))#

#color(brown)("This is an exact value so keep it in this form for now to reduce error")# #color(brown)("on final calculation.")#
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#color(blue)("To determine h")#

#h=Asin((5pi)/12)#

#=>h=3xxsin(pi/12)xxsin((5pi)/12)#

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#color(blue)("To determine area")#

#"area "= B/2xxh#

#"area "= 3/2 xx3xx sin(pi/12)xxsin((5pi)/12)#

but #sin(pi/12)xxsin((5pi)/12)=1/4#

#"area "= 3/2 xx3xx1/4#

#color(green)("area " = 1.125 " units"^2 -> 1 1/8 " units"^2)#