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 $^\circ$.
\(\displaystyle \) | \(\) | \(\displaystyle 1\) | $\quad$ degree | $\quad$ | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 60\) | $\quad$ minutes | $\quad$ | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 60 \times 60 = 3600\) | $\quad$ seconds | $\quad$ | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 {360}\) | $\quad$ full angle (by definition) | $\quad$ |
Minute
The minute (of arc) is a measurement of plane angles, symbolized by $'$.
- $1$ minute $= 60$ seconds (of arc) $= \dfrac 1 {60}$ of a degree of arc.
Second
The second (of arc) is a measurement of plane angles, symbolized by $''$.
- $1''$ is defined as $\dfrac 1 {60}$ of a minute of arc, or $\dfrac 1 {3600}$ of a degree of arc.
Radian
The radian is a measure of plane angles symbolized either by the word $\mathrm{rad}$ or without any unit.
Radians are pure numbers, as they are ratios of lengths. The addition of $\mathrm{rad}$ is merely for clarification.
$1 \ \mathrm{rad}$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius:
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.