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

1 Answer
Jan 19, 2018

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.