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