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.

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.

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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.

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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.

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