Apply the law of sines to the following: $a=\sqrt{5}, c=2 \sqrt{5}, A=30^{\circ}$. What is the value of $\sin C$ ? What is the measure of $C$ ? Based on its angle measures, what kind of triangle is triangle $A B C$ ? What is the value of $\sin C$ ? (Type an exact answer, using radicals as needed.)

Step 1 ：We are given that the sides of the triangle are \(a = \sqrt{5}\), \(c = 2 \sqrt{5}\), and the angle \(A = 30^\circ\). We are asked to find the value of \(\sin C\) and the measure of angle \(C\).

Step 2 ：We can use the law of sines, which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. This can be written as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

Step 3 ：We can rearrange the law of sines to solve for \(\sin C\): \(\sin C = \frac{c \sin A}{a}\).

Step 4 ：Substituting the given values into this equation, we find that \(\sin C \approx 1\).

Step 5 ：This suggests that the angle \(C\) is close to \(90^\circ\), but we need to calculate it to be sure.

Step 6 ：Using the inverse sine function, we find that \(C \approx 90^\circ\).

Step 7 ：Final Answer: The value of \(\sin C\) is \(\boxed{1}\) and the measure of \(C\) is \(\boxed{90^\circ}\).