Question

Transform equation: y = √x from rectangular to polar form.

Answer 1

$\bf \begin{cases} x=rcos(\theta )\\ y=rsin(\theta )\\ x^2+y^2=r^2\implies y^2=r^2-x^2 \end{cases} \\\\\\ sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)\\\\ -------------------------------\\\\ y=\sqrt{x}\implies y^2=x\implies r^2-x^2=x\implies \cfrac{r^2-x^2}{x}=1 \\\\\\ \cfrac{r^2}{x}-\cfrac{x^2}{x}=1\implies \cfrac{r^2}{x}-x=1\implies \cfrac{r^2}{rcos(\theta )}-rcos(\theta )=1$

$\bf \cfrac{r}{cos(\theta )}-rcos(\theta )=1\implies r\left[ \cfrac{1}{cos(\theta )}-cos(\theta ) \right]=1 \\\\\\ r\left[ \cfrac{1-cos^2(\theta )}{cos(\theta )} \right]=1\implies r\left[ \cfrac{sin^2(\theta )}{cos(\theta )} \right]=1\implies r=\cfrac{1}{\frac{sin^2(\theta )}{cos(\theta )}} \\\\\\ r=\cfrac{cos(\theta )}{sin^2(\theta )}$