In the plane,
after designating a pole, the origin,
and a single polar axis pointing eastward,
we can describe the location of a point
with its polar coordinates,
an ordered pair \((r,\theta)\)
where \(r\) is the length of the segment from the origin to the point
and \(\theta\) is the angle from the polar axis to the segment.
Positive \(\theta\) corresponds to counterclockwise rotation.
Altogether this constitutes the polar coordinate system.
In this context, the sine and cosine functions
should be thought as returning percentages by which the radius is scaled
as it’s projected onto the polar and normal axes.
\[
\begin{align*}
x&=r\cos(\theta)
\\ y&=r\sin(\theta)
\end{align*}
\quad
\longleftrightarrow
\quad
\begin{align*}
r&=\sqrt{x^2+y^2}
\\ \theta&=\operatorname{arctan}\Bigl(\frac{y}{x}\Bigr) \;(\text{kinda})
\end{align*}
\]
The arctangent function (tan-1)
takes a slope as input
and returns an angle between -90° and 90° as output;
if \(x\) is negative, you’ve got to think a bit about \(\theta.\)
The polar coordinates of a point are not unique.
However each point has a unique “principal” pair of polar coordinates,
where \(r \gt 0\) and \(0 \leq \theta \lt 360°.\)
Navigation on earth relies on the mathematics of polar coordinates, but uses different conventions and terminology. An object’s altitude relative to an observer is the vertical angle between that object and the observer’s local horizontal, parallel to earth. Positive altitudes are called angles of elevation (or inclination) whereas negative altitudes are called angles of depression (or declination). An object’s azimuth relative to an observer is the angle between that object and north (N) within their local horizontal. For an observer that’s moving, their heading is the direction they’re pointing/facing, whereas their track is the direction they’re actually going, and their course is the direction they intend to be going. A object’s bearing is the direction of the object from the observer described either relative to the observer’s heading or absolutely as a direction away from north, its azimuth.