# Thread: Help with cartesian equation

1. ## Help with cartesian equation

x = 2 cot t
y = 2 sin^2 t

I need to find the cartesian equation in the form y=f(x). Can somebody pls help

2. Hi

Use xē = cotēt = cosēt / sinēt = (1-sinēt)/sinēt
substitute sinēt=y/2
and finally rearrange to get y=f(x)

3. thanks you're such a lifesaver

4. Hello, threerose!

Here's another approach.
. . It should surprise/impress/terrify your professor.

$\displaystyle \begin{array}{c}x \:=\: 2\cot\theta \\ y \:=\: 2\sin^2\theta \end{array}$

Find the cartesian equation in the form $\displaystyle y\,=\,f(x)$

We have: .$\displaystyle \begin{array}{cccccccc} x \;=\;2\cot\theta &\Longrightarrow& \cot\theta \;=\; \dfrac{x}{2} &\Longrightarrow& \cot^2\theta \;=\; \dfrac{x^2}{4} & [1] \\ \\[-3mm] y \;=\; 2\sin^2\theta &\Longrightarrow& y \;=\; \dfrac{2}{\csc^2\theta} &\Longrightarrow & \csc^2\theta \;=\; \dfrac{2}{y} & [2] \end{array}$

$\displaystyle \text{Subtract }[2]\text{ - }[1]\!:\;\;\underbrace{\csc^2\!\theta - \cot^2\!\theta}_{\text{This is 1}} \;=\;\frac{2}{y} - \frac{x^2}{4}$

. . So we have: .$\displaystyle 1 \:=\:\frac{2}{y} - \frac{x^2}{4}$

Multiply by $\displaystyle 4y\!:\;\;4y \;=\;8 - x^2y \quad\Rightarrow\quad x^2y+4y \:=\:8 \quad\Rightarrow\quad y(x^2+4)\:=\:8$

Therefore: .$\displaystyle y \;=\;\frac{8}{x^2+4}$