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
namralos
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.

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