Definition:Angle/Unit
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:
Also see
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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.
