Definition:Angle/Unit

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Definition

The usual units of measurement for angle are as follows:


Degree

The degree (of arc) is a measurement of plane angles, symbolized by $\degrees$.

\(\displaystyle \) \(\) \(\displaystyle 1\) degree
\(\displaystyle \) \(=\) \(\displaystyle 60\) minutes
\(\displaystyle \) \(=\) \(\displaystyle 60 \times 60 = 3600\) seconds
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {360}\) full angle (by definition)


Minute

The minute (of arc) is a measurement of plane angles, symbolized by $'$.

\(\displaystyle \) \(\) \(\displaystyle 1\) minute
\(\displaystyle \) \(=\) \(\displaystyle 60\) seconds
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {60}\) degree of arc (by definition)


Second

The second (of arc) is a measurement of plane angles, symbolized by $''$.

\(\displaystyle \) \(\) \(\displaystyle 1\) second
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {60}\) minute of arc (by definition)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {60 \times 60} = \dfrac 1 {3600}\) degree of arc


Radian

The radian is a measure of plane angles symbolized either by the word $\radians$ or without any unit.

Radians are pure numbers, as they are ratios of lengths. The addition of $\radians$ is merely for clarification.


$1 \radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius:

Radian.png


Also see


In the context of calculus in particular, it is usual to use radians.

This is because then the derivatives of trigonometric functions, such as, for example:

$\dfrac {\mathrm d} {\mathrm d x} \sin x = \cos x$

work out without the necessity of multiplicative constants.