Radian Measure
Key Questions
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Since
180^circ =pi radians,if you want to convert
x degrees to radians, thenx times pi/180 radians,and if you want to convert
x radians to degrees, thenx times 180/pi degrees
I hope that this was helpful.
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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 of360^0 equals to2pi 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