# Convert the following polar equation into a cartesian equation. Specifically describe the graph of the equation in rectangular coordinates: r=5\sin\theta

Convert the following polar equation into a cartesian equation. Specifically describe the graph of the equation in rectangular coordinates: $r=5\mathrm{sin}\theta$
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Step 1
To convert from polar to Cartesian from.
Step 2
Given:
$r=5\mathrm{sin}\theta$
Step 3
Formula used,
$x=r\mathrm{cos}\theta \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}0$
${r}^{2}={x}^{2}+{y}^{2}$
Step 4
Simplify as,
$r=5\mathrm{sin}\theta$
Multiply by r as,
${r}^{2}=5r\mathrm{sin}\theta$

${x}^{2}+{y}^{2}-5y=0$
${x}^{2}+{y}^{2}-5y+\frac{25}{4}=\frac{25}{4}$
${x}^{2}+{\left(\frac{y-5}{2}\right)}^{2}={\left(\frac{5}{2}\right)}^{2}$
Step 5
Hence, the Cartesian form of the equation is ${x}^{2}+{\left(\frac{y-5}{2}\right)}^{2}={\left(\frac{5}{2}\right)}^{2}$ which is the equation of circle with center $\left(0,\frac{5}{2}\right)$ and radius $\frac{5}{2}$.