Definition:Angular Measure
Definition
The usual units of measurement for angle are as follows:
Degree
The degree (of angle) is a measurement of plane angles, symbolized by $\degrees$.
| \(\ds \) | \(\) | \(\ds 1\) | degree | |||||||||||
| \(\ds \) | \(=\) | \(\ds 60\) | minutes | |||||||||||
| \(\ds \) | \(=\) | \(\ds 60 \times 60 = 3600\) | seconds | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {360}\) | full angle (by definition) |
Minute
The minute (of angle) is a measurement of plane angles.
| \(\ds \) | \(\) | \(\ds 1\) | minute | |||||||||||
| \(\ds \) | \(=\) | \(\ds 60\) | seconds | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {60}\) | degree of angle (by definition) |
Second
The second (of angle) is a measurement of plane angles.
| \(\ds \) | \(\) | \(\ds 1\) | second | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {60}\) | minute of angle (by definition) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {60 \times 60} = \dfrac 1 {3600}\) | degree of angle |
Radian
The radian is the SI unit of plane angle.
It can be 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:
- $\dfrac \d {\d x} \sin x = \cos x$
work out without the necessity of multiplicative constants.
- Results about angular measure can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): angular measure
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): angular measure
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): angular measure
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): angular measure