# How do you convert r=2sin theta + cos theta into rectangular form?

Feb 25, 2017

The equation is ${\left(x - \frac{1}{2}\right)}^{2} + {\left(y - 1\right)}^{2} = \frac{5}{4}$

#### Explanation:

To convert from polar coordinates $\left(r , \theta\right)$ to rectangular coordinates $\left(x , y\right)$, we use the following equations

$x = r \cos \theta$, $\implies$, $\cos \theta = \frac{x}{r}$

$y = r \sin \theta$, $\implies$, $\sin \theta = \frac{y}{r}$

${x}^{2} + {y}^{2} = {r}^{2}$

Therefore,

$r = 2 \sin \theta + \cos \theta$

$r = 2 \cdot \frac{y}{r} + \frac{x}{r}$

${r}^{2} = 2 y + x$

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

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

${x}^{2} - x + \frac{1}{4} + {y}^{2} - 2 y + 1 = 1 + \frac{1}{4}$

${\left(x - \frac{1}{2}\right)}^{2} + {\left(y - 1\right)}^{2} = \frac{5}{4}$

This is the equation of a circle, center $\left(\frac{1}{2} , 1\right)$ and radius $= \frac{\sqrt{5}}{2}$