Polar and Rectangular Coordinates

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Section 1.2 Polar and Rectangular Coordinates

Section 1.1 introduced polar coordinates from scratch, without reference to rectangular coordinates. How do these coordinate systems compare?

Figure 1.2 redraws Figure 1.1 to show the values of the rectangular coordinates \((x,y)\) as well as the values of the polar coordinates \((r,\phi)\) for an arbitrary point \(P\text{.}\) Using the circle definition of the trigonometric functions, we see immediately that

\begin{align} x \amp= r\cos\phi ,\tag{1.2.1}\\ y \amp= r\sin\phi .\tag{1.2.2} \end{align}

These expressions can of course be inverted, yielding

\begin{align} r \amp= \sqrt{x^2+y^2} ,\tag{1.2.3}\\ \phi \amp= \tan^{-1}\left(\frac{y}{x}\right) ,\tag{1.2.4} \end{align}

which can be expressed more simply as

\begin{align} r^2 \amp= x^2 + y^2 ,\tag{1.2.5}\\ \tan\phi \amp= \frac{y}{x} .\tag{1.2.6} \end{align}

Figure 1.2. The construction of the polar coordinates (\(r\text{,}\)\(\phi\)) at an arbitrary point \(P\text{,}\) showing their relationship to rectangular coordinates (\(x\text{,}\)\(y\)).