15.4 Polar Coordinates
On the unit circle we saw that the cosine and sine functions tell us how the \( x \) and \( y \)-coordinates depend on an angular coordinate \( \ang \). We can move off the unit circle by introducing a radial coordinate \( \rad \).
Definition.
In the \( (x, y) \)-coordinate plane we may locate any point using a radius \( \rad : \Number \) and an angle \( \ang : \Number \).
\[
\begin{align}
x &= \rad \mult \cos \ang
\\
y &= \rad \mult \sin \ang
\end{align}
\]
We'll call the variables \( \rad \) and \( \ang \) polar coordinates to distinguish them from standard coordinates, \( x \) and \( y \).
To keep things simple we'll only allow positive \( \rad \)-values, \( \rad \ge 0 \), when working with polar coordinates.
Theorem.
Suppose \( \rad : \Number \) is the radial polar coordinate. Then
\[
\begin{align}
x^2 + y^2 &= \rad^2.
\end{align}
\]
Why?
This is easy to check.
\[
\begin{align}
x^2 + y^2 &= (\rad \mult \cos \ang)^2 + (\rad \mult \sin \ang)^2 = \rad^2 \mult \bigl((\cos \ang)^2 + (\sin \ang)^2\bigr) = \rad^2
\end{align}
\]
Suppose \( R : \Number \) is a constant and \( R \gt 0 \). Fixing the radial coordinate
\[
\begin{align}
&\{ {\rad = R} \}
\end{align}
\]
describes the circle with radius \( R \).
If we instead set \( \rad \) equal to zero
\[
\begin{align}
&\{ {\rad = 0} \}
\end{align}
\]
we find the origin, only.
Our graph paper looks quite different when we switch from standard coordinates to polar coordinates. Each positive radius, \( \rad \gt 0 \), gives a circle, and each angle \( \ang \) gives a ray that starts at the origin.
Example.
Consider the point \( p : \Point_{(x, y)} \) with polar coordinates
\[
\begin{align}
\rad &= 3
&
&\andSpaced
&
\ang &= \frac{1}{12} \revolution.
\end{align}
\]
Let's find the standard coordinates for this point.
\[
\begin{align}
x &= 3 \mult \cos \biggl(\frac{1}{12} \revolution\biggr) = 3 \mult \sqrt[2]{\frac{3}{4}}
\\
y &= 3 \mult \sin \biggl(\frac{1}{12} \revolution\biggr) = 3 \mult \sqrt[2]{\frac{1}{4}}
\end{align}
\]
We can locate our point in the plane using standard graph paper or polar graph paper.
