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Doing the Ambiguous Case

Given a triangle, and two sides, there is a degree of uncertainty. That is called the "Ambiguous Case."

When looking at this, we have a specific technique of solving this type of triangle.

First, look to see if it is possible. Given triangle ABC, side b = 3, and side c = 100, and measure of Angle B is 80 degrees, it must be checked whether it is possible. So, the smallest side length for B is 100 sin(80), which is 98.48, so it's impossible.

Next, if the side opposite the angle is >= that value, there is only one possible triangle. The only reason there's an ambiguous case, is you can reflect the opposite side over an imaginary altitude (hard to describe w/o pictures).

Last, if there are two possible cases, keep track of both possible cases. You know one angle, and the other two have multiple possibilities. 

De Moivre's Theorem