Polar to cartesian conversion formulas, Mathematics

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Polar to Cartesian Conversion Formulas

x = r cos Θ

y = r sin Θ

Converting from Cartesian is more or less easy.  Let's first notice the subsequent.

x2 + y2  = (r cosΘ)2 + (r sinΘ)2

= r2 cos2 Θ + r2 sin2 Θ

= r2 (cos2 Θ + sin2 Θ) = r2

This is a very helpful formula that we should keep in mind, though we are after an equation for r so let's take the square root of both sides. This provides,

r = √x2 + y2

Note: Technically we should have a plus or minus in front of the root as we know that r can be either positive or negative.  We will run along with the convention of positive r here.

Getting an equation for Θ is approximately as simple.  We'll begin with,

y/x = r sinΘ / r cos Θ = tan Θ

Taking the inverse tangent of both of the sides gives,

Θ = tan-1 (y/x)

We will require to be careful with this as inverse tangents only return values in the range - Π/2 < Θ < Π/2.  Remind that there is a second possible angle and that the second angle is provided by Θ + Π.

After that Summarizing gives the subsequent formulas for transforming from Cartesian coordinates to polar coordinates.

Cartesian to Polar Conversion Formulas

r2 = x2 + y2                                          

r = √ (x2 + y2)

Θ = tan -1 (y/x)


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