A triangle has sides A, B, and C. The angle between sides A and B is pi/6. If side C has a length of 25 and the angle between sides B and C is pi/12, what is the length of side A?

1 Answer
May 5, 2017

Explanation:

In general, the Law of Sine states that if a, b, and c are the lengths of the sides opposite angles alpha, beta, gamma (in that order), then
a/sin(alpha) = b/sin(beta) = c/sin(gamma).

We only use the Law one pair of angles at a time. In this case, we know that the angle between A and B is gamma, and the angle between B and C is alpha. Therefore,

a = unknown
c = 25
alpha = pi/12
gamma = pi/6

Using the Law:

a/sin(pi/12) = 25/sin(pi/6)

pi/6 is one of the standard angles. sin(pi/6) = 1/2.

The value of sin(pi/12) may be found using the difference formula for sine:

sin(pi/12) = sin((3pi)/12 - (2pi)/12)
= sin((3pi)/12)cos((2pi)/12)- cos((3pi)/12)sin((2pi)/12)
= sin(pi/4)cos(pi/6)- cos(pi/4)sin(pi/6)
= (sqrt2/2)(sqrt3/2) - (sqrt2/2)(1/2)
= (sqrt6 - sqrt2)/4

Solving the proportion:

a/sin(pi/12) = 25/sin(pi/6)

a = ((25)(sin(pi/12)))/sin(pi/6)

a = ((25)(sqrt6 - sqrt2)/4)/(1/2)

a = ((25)(sqrt6 - sqrt2))/2

a = (25(sqrt6 - sqrt2))/2

NOTE: If we use the half-angle formula for sine instead of the difference formula, we obtain an answer that looks different but is equal to the above.