Question

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?

Answer 1

Use the Law of Sines

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.