For an angle to to coterminal with , it must be equivalent to that angle, pointing up. The two angles that work are and .

We can visualize why works in a couple different ways. If we know that the angle pointing straight down is , we can see that the negative version of that would point straight up. We could also count clockwise around the circle three -angles, which would place us at .

Similarly, counting around the unit circle nine -angles would place us at . We could also subtract , or equivalently , to figure out where the angle is within the unit circle: 

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa. For an angle of 390° we find its positive coterminal angle by subtracting 360°. This gives us the coterminal angle of 30°. 30° and 390° both have the same initial and terminal sides.

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa. To find the negative coterminal angle of 380°, we must subtract an angle that has the same terminal and initial sides, but is larger than 380° (this ensures we get a negative coterminal angle).

To find the correct "amount of angle" to subtract, multiply 360 by multiples of 2 until you get an angle value that would give a negative angle when subtracted.

Adding the two fractions together yields

We can find angles coterminal to this by adding or subtracting multiples of .
In this case:

.

"Coterminal angles" are those angles in standard position that have a common terminal side. To find an angle coterminal to another given angle, simply add or subtract  (or ) to the given angle measure. The problem restricts the desired coterminal angles to those which lie between  and . Hence, for each given angle measure we must find the angle between  and  which is equivalent to that angle by adding or subtracting multiples of  or .

 is negative, so  is coterminal to  and lies between  and .

 is greater than , so  is coterminal to  and lies between  and .

 is greater than , so  is coterminal to  and lies between  and .

Hence, the positive angles between  and  which are coterminal to , , and  are , , , respectively.

To find the positive angle that is coterminal with the angle , must add to the giving angle. 

Therefore,

To find the negative angle that is coterminal with the angle , must subtract to the giving angle. 

Therefore,

To find the positive angle that is coterminal with the angle , must add  to the giving angle. 

Therefore,

To find the negative angle that is coterminal with the angle , must subtract  to the giving angle. 

Therefore,

Each quadrant represents a  change in radians. Therefore, an angle of  radians would pass through quadrants , , and end in quadrant . The movement of the angle is in the clockwise direction because it is negative.