# How do you convert r = 8 sin theta into cartesian form?

Dec 29, 2016

The answer is ${x}^{2} + {\left(y - 4\right)}^{2} = {4}^{2}$

#### Explanation:

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

$x = r \cos \theta$

$y = r \sin \theta$

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

Here, we have

$r = 8 \sin \theta$

$r = 8 \cdot \frac{y}{r}$

${r}^{2} = 8 y$

${x}^{2} + {y}^{2} = 8 y$

${x}^{2} + {y}^{2} - 8 y = 0$

Completing the squares

${x}^{2} + {y}^{2} - 8 y + 16 = 16$

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

This is the equation of a circle, center $\left(0 , 4\right)$ and radius $= 4$

graph{x^2+(y-4)^2=16 [-13.88, 14.6, -2.85, 11.39]}

Dec 29, 2016

${x}^{2} + {y}^{2} = 8 y$

#### Explanation:

we understand that;

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

from the question,

$r = 8 \sin \theta$
we can multiply with r, then

$r \cdot r = r \cdot 8 \sin \theta$
${r}^{2} = 8 \left(r \sin \theta\right) w h e r e , y = r \sin \theta$
${r}^{2} = 8 y$

therefore from @ the answer is
${x}^{2} + {y}^{2} = 8 y$