A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/2# and the angle between sides B and C is #pi/12#. If side B has a length of 7, what is the area of the triangle?

1 Answer
Zor Shekhtman
Feb 4, 2016

#S = 49(1-sqrt(3)/2)~=6.5648#

Explanation:

It is a right triangle with catheti #A# and #B# and hypotenuse #C#.
Knowing the length of #B=7# and an angle between #B# and #C# equaled #pi/12#, we can calculate #A# as follows.

Since #A/B = tan(pi/12)#,
it follows that #A = B*tan(pi/12)#

Area of a triangle is
#S =1/2(A*B) = 1/2B^2tan(pi/12)#

The latter can be simplified using a formula
#tan(phi) = (1-cos(2phi)) / sin(2phi)#
from which follows:
#tan(pi/12) = (1-cos(pi/6)) / sin(pi/6) = [1-sqrt(3)/2]/(1/2) = 2-sqrt(3)#

Therefore, the area of a triangle is
#S = (1/2)7^2(2-sqrt(3))=49(1-sqrt(3)/2)#

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