## Coterminal angles

Consider the angle , in standard position.

Now consider the angle . We can think of this angle as a full rotation , plus an additional 30 degrees.

Notice that looks the same as . Formally, we say that the angles share the same terminal side. Therefore we call the angles co-terminal. Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another , we get the angle . Or, if we create the angle in the negative direction (clockwise), we get the angle . Because we can rotate in either direction, and we can rotate as many times as we want, we can keep generating angles that are co-terminal with .

Example 3. Which angles are co-terminal with ?

a.

b.

c.

d.

Solution: b. and c. are co-terminal with .

Notice that terminal side of the first angle, , is in the quadrant. The last angle, is in the quadrant. Therefore neither angle is co-terminal with .

Now consider . This is a full rotation, plus an additional . So this angle is co-terminal with . The angle can be generated by rotating clockwise. To determine where the terminal side is, it can be helpful to use quadrantal angles as markers. For example, if you rotate clockwise (for a total of ), the terminal side of the angle is on the positive axis. For a total clockwise rotation of , we have more to rotate. This puts the terminal side of the angle at the same position as .