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Section 12.12 : Cylindrical Coordinates

5. Convert the following equation written in Cylindrical coordinates into an equation in Cartesian coordinates.

\[4\sin \left( \theta \right) - 2\cos \left( \theta \right) = \frac{r}{z}\]

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Start Solution

There really isn’t a whole lot to do here. All we need to do is to use the following \(x\) and \(y\) polar conversion formulas in the equation where (and if) possible.

\[x = r\cos \theta \hspace{0.5in}y = r\sin \theta \hspace{0.5in}{r^2} = {x^2} + {y^2}\] Show Step 2

To make the conversion a little easier let’s multiply the equation by \(r\) to get,

\[4r\sin \left( \theta \right) - 2r\cos \left( \theta \right) = \frac{{{r^2}}}{z}\] Show Step 3

Now let’s use the formulas from Step 1 to convert the equation into Cartesian coordinates.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{4y - 2x = \frac{{{x^2} + {y^2}}}{z}}}\]