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 $^\circ$.

\(\displaystyle \) \(\) \(\displaystyle 1\) $\quad$ 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:

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.