Recall that when an angle is drawn in the standard position as above, only the terminal sides (BA, BD) varies, since the initial side (BC) remains fixed along the positive x-axis.
If two angles are drawn, they are coterminal if both their terminal sides are in the same place - that is, they lie on top of each other. In the figure above, drag A or D until this happens.
If the angles are the same, say both 60°, they are obviously coterminal. But the angles can have different measures and still be coterminal. In the figure above, rotate A around counterclockwise past 360° until it lies on top of DB. One angle (DBC) has a measure of 72°, and the other (ABC) has a measure of 432°, but they are coterminal because their terminal sides are in the same position. If you drag AB around twice you find another coterminal angle and so on. There are an infinite number of times you can do this on either angle.
In the figure above, drag D around the origin counterclockwise so the angle is greater than 360°. Now drag point A around in the opposite direction creating a negative angle. Keep going until angle DBC is coterminal with ABC. You can see that a negative angle can be coterminal with a positive one.
You can sketch the angles and often tell just form looking at them if they are coterminal. Otherwise, for each angle do the following:
In trigonometry we use the functions of angles like sin, cos and tan. It turns out that angles that are coterminal have the same value for these functions. For example, 30°, 390° and -330° are coterminal, and so sin30°, sin390° and sin(-330°) and all have the same value (0.5).