MIT grad shows how to convert from radians to degrees, convert from degrees to radians, and explains what radians and degrees mean. To skip ahead: 1) For converting from DEGREES TO RADIANS, skip to time 4:50. 2) For converting from RADIANS TO DEGREES, skip to time 8:11. 3) For CONVERTING A DECIMAL number of radians into degrees, skip to time 10:50. Nancy formerly of MathBFF explains the steps.
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Radians and degrees are just two different ways of expressing an angle, and they have different units. Angle measures written in degrees have a little superscript circle symbol after the number. Angle measures written in radians do not have the degree circle symbol and may or may not have the units "radians" or the abbreviation "rad." listed after the number.
1) TO CONVERT FROM DEGREES TO RADIANS: Multiply the number of degrees by the conversion factor PI OVER 180. For instance, to convert from 200 degrees into radians, you would multiply 200 degrees by the fraction quantity pi radians over 180 degrees. Since you are starting with degrees, and your conversion factor has degrees in the denominator, the degrees units will go away, or effectively cancel, and you will be left with the radian units you want. 200 times pi, divided by 180, simplifies to the quantity 10 pi over 9, which is now in radians. You can leave the symbol pi in your final answer.
2) TO CONVERT FROM RADIANS TO DEGREES: Multiply the number of radians by the conversion factor 180 OVER PI. For instance, to convert from 2 pi over 9 radians into degrees, you would multiply 2 pi over 9 by the fraction quantity 180 degrees over pi radians. Since you are starting with degrees, and your conversion factor has radians in the denominator, the radians units go away, or cancel out, and you will be left with the degree units you want. Since you are also starting with a pi in your angle measure, and the conversion factor has a pi in the denominator, the pi symbols will cancel out and you're left with just 2 times 180 over 9. 2 times 180 is 360, and 360 divided by 9 simplifies to 40, so your final answer is 40 degrees.
What is a radian? Technically, the definition is that if you had a circle with a radius r, an arc length distance of r around the circle would create an inside, central angle that is one radian.
How many degrees are in a circle? There are 360 degrees total in one full circle. Starting from zero degrees at the right-most point on the circle and moving counter-clockwise, one quarter of the way through a circle is 90 degrees, one half of the circle is 180 degrees, three-quarters of the way is 270 degrees, and one full circle is 360 degrees and brings you all the way back to your starting 0 degree point.
How many radians are in a circle? There are 2 pi radians total in one full circle. Starting from zero radians at the right-most point on the circle moving counter-clockwise, one quarter of the way through a circle is pi over 2 radians, one half of the circle is pi radians, three-quarters of the way is 3 pi over 2 radians, and one full circle is 2 pi radians and brings you back to your starting 0 radian point.
Notice that 360 degrees is equivalent to, or the same as, 2 pi. Since 360 degrees equals 2 pi, you could divide both sides of this equation by 2 and find that 180 degrees equals pi. You will use this equivalence, the fact that 180 degrees equals pi, to convert from degrees to radians, or from radians to degrees.
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