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# Prove that for every revolution it is equivalent to 360degrees or 2pi rad.

I'm very curious about revolution. Why 1 revolution = 360degrees or 2pi rad? Is there proved that 1turn is 360?

Note by John Aries Sarza
4 years, 3 months ago

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< Source: Wolfram Alpha, www.wolframalpha.com >

The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1. A full angle is therefore 2pi radians, so there are 360° per 2pi radians, equal to 180°/pi or 57.29577951°/radian. Similarly, a right angle is pi/2 radians and a straight angle is pi radians. Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms, e.g., d/(dx) sinx = cosx for x measured in radians. Unless stated otherwise, all angular quantities considered in this work are assumed to be specified in radians. · 4 years, 3 months ago

The very definition of degree is chosen to suit the fact that 1 revolution $$= 360^0$$.

Note that units are not defined naturally, humans select the suitable units for ease of calculations. For example the definition of degree is as follows.

Divide the complete revolution of a straight line into 360 equal parts. Then each equal part of the revolution will be equivalent to $$1^0$$. So as we can see there is no proof for 1 revolution $$= 360^0$$, the unit 'degree' is chosen in such a way that it comes true.

Now the unit radian is defined as follows.

The angle which an arc on a circle with length equal to the radius of the circle subtends at the center is called 1 radian.

Now of a circle with radius $$r$$, from the definition, an arc with length $$r$$ will subtend an angle 1 radian at the center. Thus an arc of length $$2\pi r$$ subtends angle $$2\pi r$$ at the center. But if you notice, an arc of length $$2\pi r$$ is actually the circumference of the circle, so it denotes a complete revolution. Thus, 1 revolution= $$2\pi r$$ radians. · 4 years, 3 months ago