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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.
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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.
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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. 3 different angles

  3. Express each of the following angle measures as an equivalent number of radians between 0 and 2π.
    1. 27 radians

    2. 42.3 radians

    3. $\sqrt{277}$ radians

    4. -36 radians

    5. 4297 radians

    6. -295 radians
  4. 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?