Given an “origin” point \(O,\) and two other points \(A\) and \(B,\)
the angle \(\angle AOB\) is the rotation
from the initial side \(\overrightarrow{OA}\)
to the terminal side \(\overrightarrow{OB}.\)
The measure of an angle is the amount of that rotation.
Conventionally, angles swept out counterclockwise are positive
whereas angles swept out clockwise are negative.
In the \(xy\)-plane this means positive angles are swept
from the positive \(x\)-axis towards the positive \(y\)-axis,
a convention called the right-hand rule.
Two angles are coterminal
if their initial and terminal sides coincide.
Coterminal angles differ only by full rotations.
The principal angle corresponding to an angle
is the smallest positive angle coterminal to it;
the coterminal angle that is less than one rotation.
(This is the same as every arclength having a “reference number”.)
There a multiple ways to measure angles.
One convention is to measure an angle
by the length of the arc subtending that angle in radii.
I.e. one full rotation corresponds to an angle of \(2\pi.\)
This is known as radian measure,
and aligns with the “unit circle” interpretation of trigonometric functions.
Another convention is to declare one full rotation to be 360°
and measure angles in how many degrees (1°) it spans.
When measuring in degrees there are two conventions
for calculating fractions of a degree:
either literally as a decimal number of degrees,
or in terms of minutes (1′) and seconds (1″)
where 1° equals 60′, and 1′ equals 60″.
E.g. 12.58222° ≈ 12°34′56″.
Principle angles, when measured in radians,
must be between 0 and \(2\pi,\)
and when measured in degrees must be between 0° and 360°.
Radian measure lends itself to simpler geometry. For an angle of radian measure \(\theta\) within a sector with initial and terminal side lengths \(r,\) the length of the arc subtending the angle will be \(r\theta\) and the area of the sector will be \(\frac{1}{2}r^2\theta.\)