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. Move your cursor away from the animation to make the control bar
disappear.

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.
Move your cursor away from the animation to make the control bar
disappear.

Exercise Set 2.4

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.

Express each of the following angle measures as an equivalent number of radians between 0 and 2π.

27 radians

42.3 radians

$\sqrt{277}$ radians

-36 radians

4297 radians

-295 radians

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?