A triangle has sides A, B, and C. The angle between sides A and B is pi/8. If side C has a length of 32 and the angle between sides B and C is pi/12, what is the length of side A?

1 Answer
Zor Shekhtman
Mar 16, 2016

A=32sqrt(2-sqrt(3))/sqrt(2-sqrt(2))

Explanation:

Angle measured pi/12 lies across side A.

Angle measured pi/8 lies across side C=32.

Using the Law of Sines, A/sin(pi/12)=32/sin(pi/8)

from which follows that A=32sin(pi/12)/sin(pi/8)

Let's determine the values of these two sines.
We will use the following trigonometric identities:
cos(2x)=cos^2(x)-sin^2(x)=1-2sin^2(x)
and, hence,
sin^2(x)=(1-cos(2x))/2

Using the above,

sin(pi/12) = sqrt(sin^2(pi/12)) = sqrt((1-cos(pi/6))/2)=
=sqrt((2-sqrt(3))/4)=1/2sqrt(2-sqrt(3))

sin(pi/8) = sqrt(sin^2(pi/8)) = sqrt((1-cos(pi/4))/2) =
= sqrt((2-sqrt(2))/4)=1/2sqrt(2-sqrt(2))

Therefore,
A=32sqrt(2-sqrt(3))/sqrt(2-sqrt(2))

Related questions
See all questions in The Law of Sines
Impact of this question
1477 views around the world
You can reuse this answer
Creative Commons License