Radian Measure
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Key Questions

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# 
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:
Throug the equivalence
#pi=180Âº# and de direct proportionality of arcs and angles. See examplesExplanation:
If we have 45Âº degrees convert to radians is easy
#pi/x=180/45# . Then#x=(45pi)/180=pi/4# . So#45Âº=pi/4 rad# By the other hand, if we have
#pi/3# rads, to convert to degrees, we use the same equivalence#pi/(pi/3)=180/y# . Trasposing terms#y=(180pi)/(3pi)=180/3=60Âº# 
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