This is a rather basic problem that I am lacking a good explanation for. Suppose you have scalene triangle $ABC$ that you want to solve, with $A=61^\circ$, $b=4$ and $c=6$.
First, you must use Law of Cosines because you know two sides and one angle. Doing this, you can solve for the third side $c$, and find that $c\approx 5.3599569$.
At this point, you know all three sides of the triangle, and one angle, so we are able to use both Law of Sines and Law of Cosines.
If we use Law of Sines to find angle $C$, we will have:
$\sin(C)\approx .9790597793$, so $C\approx78.254$. However, since we do not yet know the third angle of the triangle, it is also possible that $C$ equal the supplement of the angle we just found, so $C_2\approx 101.746$. We check to see that this is a viable measure, and it is because $C_2+A<180^\circ$, so there are two triangles that correspond with this value of sine.
However, if we instead used the Law of Cosines, we would find that $C\approx 78.254$. So my question is, does Law of Cosines "miss" an answer that Law of Sines finds, or does this result from Law of Cosines show that $C_2$ is not a valid answer, even though it otherwise checks out?
Thanks in advance.