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 counterclockwise 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?