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

It would be easy to think that all angle measures θ must be between 0 and 2π 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. Its distance is not limited to 2π times the track’s radius, so the arc length of the central angle that subtends it will not stay between 0 and and 2π radians. This is illustrated in Figure 2.4.1.

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.

Likewise, if the car travels clockwise, the arc length of the central angle that subtends it will not stay between 0 and -2π radians. This is illustrated in Figure 2.4.2.

Figure 2.4.2. All angle measures can be expressed as $x + 2πn$ for some
value of x, $0 ≤ x < 2π$, 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?