# How do you convert the polar equation r=3sintheta into rectangular form?

Oct 5, 2016

${x}^{2} + {\left(y - \frac{3}{2}\right)}^{2} = \frac{9}{4}$

#### Explanation:

$x = r \cos \left(\theta\right)$
$y = r \sin \left(\theta\right)$
${r}^{2} = {x}^{2} + {y}^{2}$

Multiply the equation by $r$

$r \cdot r = 3 r \sin \left(\theta\right)$

Simplify

${r}^{2} = 3 r \sin \left(\theta\right)$

Make appropriate substitutions

${x}^{2} + {y}^{2} = 3 y$

Gather all of the terms to the same side

${x}^{2} + {y}^{2} - 3 y = 0$

Complete square using the coefficient of $y$ variable

${\left(- \frac{3}{2}\right)}^{2} = \frac{9}{4}$

Add $\frac{9}{4}$ to both sides the equation to keep it balanced. The constant $\frac{9}{4}$ allows you make a perfect square trinomial.

${x}^{2} + {y}^{2} - 3 y + \frac{9}{4} = \frac{9}{4}$

Rewrite

${x}^{2} + {\left(y - \frac{3}{2}\right)}^{2} = \frac{9}{4}$

Check out a tutorial on converting an equation from polar to rectangular

Check out a tutorial on completing the square graphically

Check out a tutorial on completing the square analytically