Question

How do you determine whether `triangle ABC` has no, one, or two solutions given `A=30^circ, a=3, b=6`?

Answer 1

One triangle.

Explanation:

In this particular case we're given `A=30^circ` and `a=3`, which is the side opposite `A`. Since `b=6=2a` we actually know that we're dealing with a `30^circ`-`60^circ`-`90^circ` triangle because of the ratio of sides: `x-xsqrt(3)-2x`. In this case there is exactly one triangle and we don't really need to use the Law of Sines.

If you want to use the law of sines, though, calculate `6*sin(30^circ) = 6(1/2)=3` which is exactly the length of `a`, the side opposite the given angle, which means we have one right triangle solution.