# How do I convert the polar equation r=10 sin theta to its Cartesian equivalent?

Oct 2, 2014

Cartesian coordinates, also known as Rectangular coordinates, are defined in terms of $x$ and $y$. So, for this problem $\theta$ has to be eliminated/converted using basic foundations described by the unit circle and right triangle trigonometry.

$r = 10 \sin \left(\theta\right)$

Remember that ...

$x = r \cdot \cos \left(\theta\right)$

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

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

Multiply both sides of the equation by $r$

$r \cdot r = 10 r \cdot \sin \left(\theta\right)$

${r}^{2} = 10 r \cdot \sin \left(\theta\right)$

${x}^{2} + {y}^{2} = 10 r \cdot \sin \left(\theta\right)$

Use the fact that $y = r \cdot \sin \left(\theta\right)$ to make a substitution.

${x}^{2} + {y}^{2} = 10 y$

${x}^{2} + {y}^{2} - 10 y = 0$

The above equation is the Cartesian/Rectangular coordinate equivalent to the given Polar equation.