polar to cartesian conversion formulasx r cos

Question

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)