Question

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

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asked 2021-04-03
Convert the following polar equation into a cartesian equation. Specifically describe the graph of the equation in rectangular coordinates: \(\displaystyle{r}={5}{\sin{\theta}}\)

Answers (1)

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 \(\displaystyle{\frac{{{5}}}{{{2}}}}\).

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