A triangle has sides A, B, and C. The angle between sides A and B is #pi/3# and the angle between sides B and C is #pi/12#. If side B has a length of 1, what is the area of the triangle?

1 Answer
Jan 19, 2016

#A~~ 0.116#

Explanation:

Sketch
The area of a triangle #A = 1/2*Base*height#
In this case the base #B=1#
If we name the two portions of the base on either side of the perpendicular #h# as #x# and #y#, then #B = x+y = 1#

We then use the trigonometric relations
#h/x = tan(pi/12)#
#h/y =tan(pi/3)#

We now have three equations and three unknowns so we can solve to find #h#
#x = h/tan(pi/12)#
#y = h/tan(pi/3)#
#:.h/tan(pi/12) + h/tan(pi/3) = 1#

#h(tan(pi/3) + tan(pi/12))/(tan(pi/12)tan(pi/3)) = 1#

#:.h = (tan(pi/12)tan(pi/3))/(tan(pi/3) + tan(pi/12))#

#A = 1/2*1*(tan(pi/12)tan(pi/3))/(tan(pi/3) + tan(pi/12))#

#A~~( 0.268*1.732)/(2(1.732 + 0.268))#

#A~~ 0.464/4 ~~ 0.116#