# 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

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

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.