State whether the given measurements determine zero, one, or two triangles.. c = 30°, a = 32, c = 16

Sin(A) / a = Sin(C)/c
Sin(A) = a * Sin(C)/c
Sin(A) = 32 * Sin(30)/16
Sin(A) = 32 * 0.03125
Sin(A) = 1

A = 90° 

By the law of sines, angle A must be 90 degrees. That makes side a the hypotenuse of a 30-60-90 triangle. There is only one triangle. The third side is 16√3, and the third angle 60°.