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Section 2.4 Angle Measures Greater than 2π or less than -2π

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).

Figure 2.4.1. Angle measures can be greater than 2π or less than –2π.
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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).

Figure 2.4.2. All angle measures can be expressed as $\theta+2n\pi$ for some value of $\theta,\,0\le\theta\le 2\pi$ and some integer n.
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Exercise Set 2.4

1. Visually estimate the radian measure of each angle in four ways: clockwise from B to C, counterclockwise from C to B, clockwise from C to B, and counterclockwise from B to C.

2. Express each of the following angle measures as an equivalent number of radians between 0 and 2π.
3. $\sqrt{277}$ radians