Coterminal angles

From co -"together", terminal -"end position"

Definition: Two angles are coterminal if they are drawn in the standard position and both have their terminal sides in the same location.
Try this: Adjust the angle below by dragging point A or D multiple times around B, the origin, and note when the angles ABC and DBC are coterminal.

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.

Either or both angles can be negative

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.

How to tell if two angles are coterminal.

You can sketch the angles and often tell just form looking at them if they are coterminal. Otherwise, for each angle do the following:

  • If the angle is positive, keep subtracting 360 from it until the result is between 0 and +360. (In radians, 360° = 2π radians)
  • If the angle is negative, keep adding 360 until the result is between 0 and +360.
If the result is the same for both angles, they are coterminal.

Why is this important?

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).

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus