# Polar coordinates to Cartesian coordinates

#### nm321

Convert the system in the polar coordinates r'=(1-r), theta' =sin(theta/2)^2 into cartesian coordinates.

#### chiro

MHF Helper
Hey nm321.

Can you show us your attempts?

#### nm321

This is all I have so far. I'm stuck

#### HallsofIvy

MHF Helper
You have a linear transformation from one polar coordinate system to another: $$\displaystyle r'= 1- r$$ and $$\displaystyle \theta'= sin^2(\theta/2)$$. I would start by using the "half angle formula":
$$\displaystyle sin(\theta/2)= \sqrt{(1/2)(1- cos(\theta)}$$ so that $$\displaystyle \theta'= sin^2(\theta/2)= \frac{1}{2}(1- cos(\theta)$$. Now use the fact that $$\displaystyle \theta= arctan\left(\frac{y}{x}\right)$$, $$\displaystyle r= \sqrt{x^2+ y^2}$$, $$\displaystyle \theta'= arctan\left(\frac{y'}{x'}\right)$$, and $$\displaystyle r'= \sqrt{x'^2+ y'^2}$$

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