Polar coordinates to Cartesian coordinates

Oct 2013
8
0
Boston, Massachusetts
Convert the system in the polar coordinates r'=(1-r), theta' =sin(theta/2)^2 into cartesian coordinates.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey nm321.

Can you show us your attempts?
 
Oct 2013
8
0
Boston, Massachusetts
This is all I have so far. I'm stuck :(
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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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