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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