Polar Bear Math

Converting Between Coordinate Systems

When we moved triangle park to the north pole, we cheated a bit and moved the corners so they'd land on the polar bear's street corners. The conversion between rectangular and polar coordinates does not come out exactly. The actual numbers came out:

(-2, 1)2.23607\(\angle\)153.43495
(1, 3)3.16228\(\angle\)71.56505
(3, -3)4.24264\(\angle\)315

Converting between rectangular and polar coordinates requires trigonometric functions. Usually, when polar graphing is used the data that is displayed is naturally based on angles and does not need to be converted back and forth to rectangular coordinates.

If you do need to convert between coordinate systems, here are the formulas to use.

Polar to Rectangular Rectangular to Polar
\(\begin{align} x &= r cos \theta \\ y &= r sin \theta \end{align}\) \(\begin{align} r &= \sqrt{x^2+y^2} \\ \theta &= tan^{-1} \frac{y}{x} \end{align}\)

Computing the \(\theta\) value is a little tricky because there are some special cases:

When \(x = 0\) then \(\theta = \begin{cases} 90^\circ, & y>0 \\ 0^\circ, & y=0 \\ 270^\circ, & y<0 \end{cases} \)
When \(x \ne 0\) then \(\theta = \begin{cases} tan^{-1}(y/x), & x>0, & y \ge 0 \\ tan^{-1}(y/x)+360^\circ, & x>0, & y < 0 \\ tan^{-1}(y/x)+180^\circ, & x<0 \end{cases} \)

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Plotting Functions on Polar Graphs
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