# How do you graph r=-2sintheta?

Oct 14, 2016

The way that you graph this is, set your compass to a radius of 1, put the center point at $\left(0 , - 1\right)$, and draw a circle.

#### Explanation:

Multiply both sides by r:

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

Substitute ${x}^{2} + {y}^{2}$ for ${r}^{2}$ and $y$ for $r \sin \left(\theta\right)$

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

Write ${x}^{2} a s {\left(x - 0\right)}^{2}$:

${\left(x - 0\right)}^{2} + {y}^{2} = - 2 y$

Add $2 y + {k}^{2}$ to both sides:

${\left(x - 0\right)}^{2} + {y}^{2} + 2 y + {k}^{2} = {k}^{2}$

Using the pattern ${\left(y - k\right)}^{2} = {x}^{2} - 2 k y + {k}^{2}$ we observe that we can use the 2y term to find the value of $k \mathmr{and} {k}^{2}$:

$- 2 k y = 2 y$

$k = - 1 \mathmr{and} {k}^{2} = 1$

Substitute this into the equation:

${\left(x - 0\right)}^{2} + {y}^{2} + 2 y + 1 = 1$

This means y terms on the left side can be written as ${\left(y - - 1\right)}^{2}$:

${\left(x - 0\right)}^{2} + {\left(y - - 1\right)}^{2} = {1}^{2}$

The way that you graph this is, set your compass to a radius of 1, put the center point at $\left(0 , - 1\right)$, and draw a circle.