Find all possible values of angle C

Question

In ΔABC, AB=5, BC=3, and angle BAC=20°. Find all possible values of angle C (ambiguous case)

Answer 1

Let side AB = c and BC = a, the name of the opposite angle to that side. Since m∠A = 20°, then c · sin A is the height of the triangle.  Since c > a > c · sin A, there are two possible solutions.  Use law of sines: a / sin A = c / sin C or alternatively sin C = c · sin A / a.  (Note, capital letters refer to angles). 

 

Now when you use arcsin, your calculator only gives you Principal values of sine (between -90° and 90°).  If you make a sketch of the problem, you will notice that the possible answers to C are in fact supplementary angles.  This is also validated by the identity that sin(θ) = sin(180-θ).