Finding Cartesian Equations for Curves Given in Polar Coordinates

Typically a curve is given in polar coordinateswithas a function ofIt is often quite simple to write this in cartesian coordinatesby making the substitutionsand simplifying the resulting expression.

Example:

On substituting these, the equation becomes

Subtract the terms on the right hand side to give

We can complete the square for both the's and's to giveThis is the equation of a circle with centreand radius 2. Note thatsatisfies the cartesian equation so lies on the curve.

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