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Radian Measure

Key Questions

  • Since

    #180^circ =pi# radians,

    if you want to convert #x# degrees to radians, then

    #x times pi/180# radians,

    and if you want to convert #x# radians to degrees, then

    #x times 180/pi# degrees


    I hope that this was helpful.

  • Answer:

    #"see explanation"#

    Explanation:

    #"to convert from "color(blue)"radians to degrees"#

    #color(red)(bar(ul(|color(white)color(black)("degrees "="radians "xx180^@/pi)color(white)(2/2)|)))#

    #"to convert from "color(blue)"degrees to radians"#

    #color(red)(bar(ul(|color(white)(2/2)color(black)("radians "="degrees "xxpi/180^@)color(white)(2/2)|)))#

  • Imagine a circle and a central angle in it. If the length of an arc that this angle cuts off the circle equals to its radius, then, by definition, this angle's measure is 1 radian. If an angle is twice as big, the arc it cuts off the circle will be twice as long and the measure of this angle will be 2 radians. So, the ratio between an arc and a radius is a measure of a central angle in radians.

    For this definition of the angle's measure in radians to be logically correct, it must be independent of a circle.
    Indeed, if we increase the radius while leaving the central angle the same, the bigger arc that our angle cuts from a bigger circle will still be in the same proportion to a bigger radius because of similarity, and our measure of an angle will be the same and independent of a circle.

    Since the length of a circumference of a circle equals to its radius multiplied by #2pi#, the full angle of #360^0# equals to #2pi# radians.

    From this we can derive other equivalencies between degrees and radians:

    #30^0=pi/6#
    #45^0=pi/4#
    #60^0=pi/3#
    #90^0=pi/2#
    #180^0=pi#
    #270^0=3pi/2#
    #360^0=2pi#

Questions