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It is natural to think that any angle measure $\theta$ must be between 0 and $2\pi$ if going in the positive direction or between 0 and –2π if going in the negative direction. But arc lengths cannot be restrained to those measures.
A race car traveling counter-clockwise on a circular track is like a point moving around a circle. The car is not limited to traveling around the track once. So the arc length of the central angle it subtends will not stay between 0 and and $2\pi$ radians (Figure 2.4.1, left).
Similarly, a car traveling this track clockwise is not limited to going around the track once. So the arc length of the central angle it subtends will not stay between 0 and and $-2\pi$ radians (Figure 2.4.1, right).
Also, if two cars travel the track starting from the same place, one travels $\theta$ radii and the other travels $\theta+2\pi$ radii, they will be at the same position on the track.
This means that angle measures of $\theta$ radii, $\theta+2\pi$ radii, $\theta+4\pi$ radii, $\theta+6\pi$ radii, etc. are equivalent in regard to where on the circle the angle's ray intersects the circle (Figure 2.4.2).
A race car started at 4.2 radians from due north. During the race it moved +288 radians. How many radians from due north did it finish, measured in a positive direction? How many radians from due north did it finish, measured in a negative direction?