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

Equations
Convert the following polar equation into a cartesian equation. Specifically describe the graph of the equation in rectangular coordinates: $$\displaystyle{r}={5}{\sin{\theta}}$$

2021-04-05

Step 1
To convert from polar to Cartesian from.
Step 2
Given:
$$\displaystyle{r}={5}{\sin{\theta}}$$
Step 3
Formula used,
$$\displaystyle{x}={r}{\cos{\theta}}{\quad\text{and}\quad}{y}={r}{\sin{{0}}}$$
$$\displaystyle{r}^{{{2}}}={x}^{{{2}}}+{y}^{{{2}}}$$
Step 4
Simplify as,
$$\displaystyle{r}={5}{\sin{\theta}}$$
Multiply by r as,
$$\displaystyle{r}^{{{2}}}={5}{r}{\sin{\theta}}$$
$$\displaystyle{x}^{{{2}}}+{y}^{{{2}}}={5}{\left({r}{\sin{\theta}}\right)}\ \ \ {\left({sin{{c}}}{e},{r}^{{{2}}}={x}^{{{2}}}+{y}^{{{2}}}\right)}$$
$$\displaystyle{x}^{{{2}}}+{y}^{{{2}}}={5}{y}\ \ \ {\left({sin{{c}}}{e},{y}={\left({r}{\sin{\theta}}\right)}\right)}$$
$$\displaystyle{x}^{{{2}}}+{y}^{{{2}}}-{5}{y}={0}$$
$$\displaystyle{x}^{{{2}}}+{y}^{{{2}}}-{5}{y}+{\frac{{{25}}}{{{4}}}}={\frac{{{25}}}{{{4}}}}$$
$$\displaystyle{x}^{{{2}}}+{\left({\frac{{{y}-{5}}}{{{2}}}}\right)}^{{{2}}}={\left({\frac{{{5}}}{{{2}}}}\right)}^{{{2}}}$$
Step 5
Hence, the Cartesian form of the equation is $$\displaystyle{x}^{{{2}}}+{\left({\frac{{{y}-{5}}}{{{2}}}}\right)}^{{{2}}}={\left({\frac{{{5}}}{{{2}}}}\right)}^{{{2}}}$$ which is the equation of circle with center $$\displaystyle{\left({0},{\frac{{{5}}}{{{2}}}}\right)}$$ and radius $$\displaystyle{\frac{{{5}}}{{{2}}}}$$.