How do I prove that another triangle might be possible for certain measurements?

Question

How do I prove that another triangle might be possible for certain measurements. For example, if the measurements given are c = 5, b = 6, and angle C = 40°, and I solved the remaining measurements which are angle A = 89.5°, angle B = 50.5°, and a = 7.78. How do I prove that another triangle might be possible with the given measurements? How do I solve the other possible measurement of angle A?

Answer 1

Solving for $\angle B$ means finding an inverse sine, using $\frac{\large \sin C}{\large c}=\frac{\large \sin B}{\large b}$. There are two solutions: the $50.5°$ solution you give (which I haven't checked), and symmetrically about $90°$, the corresponding other solution would be $129.5°$. The alternative values of $\angle A$ and $a$ follow from that .