Coterminal angles

Consider the angle 30^\circ, in standard position.

Now consider the angle 390^\circ. We can think of this angle as a full rotation (360^\circ), plus an additional 30 degrees.

Notice that 390^\circ looks the same as 30^\circ. 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 360^\circ, we get the angle 750^\circ. Or, if we create the angle in the negative direction (clockwise), we get the angle -330^\circ. 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 30^\circ.

Example 3. Which angles are co-terminal with 45^\circ ?

a. -45^\circ

b. 405^\circ

c. -315^\circ

d. 135^\circ

Solution: b. 405^\circ and c. -315^\circ are co-terminal with 45^\circ.

Notice that terminal side of the first angle, -45^\circ, is in the 4^{th} quadrant. The last angle, 135^\circ is in the 2^{nd} quadrant. Therefore neither angle is co-terminal with 45^\circ.

Now consider  405^\circ. This is a full rotation, plus an additional 45\;\mathrm{degrees}. So this angle is co-terminal with 45^\circ. The angle -315^\circ 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 90\;\mathrm{degrees}\ 3\;\mathrm{times} (for a total of 270\;\mathrm{degrees}), the terminal side of the angle is on the positive y-axis. For a total clockwise rotation of 315\;\mathrm{degrees}, we have 315-270 = 45\;\mathrm{degrees} more to rotate. This puts the terminal side of the angle at the same position as 45^\circ.