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15 August, 10:32

Explain the relationship (s) among angle measure in degrees, angle measure in radians, and arc length

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  1. 15 August, 10:45
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    Let's briefly imagine some new, simple measure for seeing where we are on the circumference a circle. We'll call this unit "rotations," and we'll define it like this:

    Let r be any number. r describes the number of rotations we've made around the circle. If r = 1, we've gone all the way around; if r = 1/2, we've gone half way around, if r = 1/4, we've gone a quarter of the way around, etc.

    Once we've established that unit, we can use it to establish a link between degrees, radians, and arc length.

    1 full rotation corresponds to:

    - 360°

    - 2π radians

    - 2πr in arc-length (where r is the radius of the circle)

    If we wanted to find unit conversions between each, we could just set up some equalities between the three:

    Radians → degrees:

    2π rad = 360°

    1 rad = (180/π) °

    Degrees → radians

    360° = 2π rad

    1° = π/180 rad

    Radians → arc-length

    2π rad = 2πr arc-length

    1 rad = r arc-length

    Arc-length → radians

    2πr arc-length = 2π rad

    1 arc-length = 1/r rad

    Arc-length → degrees

    2πr arc-length = 360°

    1 = 180/πr arc-length

    Degrees → arc-length

    360° = 2πr arc-length

    1° = πr/180 arc-length
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