# How do you convert r = 2sintheta - costheta into cartesian form?

Jun 9, 2016

${x}^{2} + {y}^{2} - 2 y + x = 0$

#### Explanation:

The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ is given by

$x = r \cos \theta$, $y = r \sin \theta$ and ${r}^{2} = {x}^{2} + {y}^{2}$

As $r = 2 \sin \theta - \cos \theta$ can also be written as (by multiplying each term by $r$)

${r}^{2} = 2 r \sin \theta - r \cos \theta$ or

${x}^{2} + {y}^{2} = 2 y - x$ or

${x}^{2} + {y}^{2} - 2 y + x = 0$

This is the equation of a circle whose graph is as given below.

graph{x^2+y^2-2y+x=0 [-2.97, 2.03, -0.23, 2.27]}