Question

The equivalent polar equation for the given rectangular - coordinate equation.
\(\displaystyle{y}=\ -{3}\)

Answer 1

Concepts used: Conversion formula for coordinate systems are given as:
a) From polar to rectangular:
\(\displaystyle{x}={r}{\cos{\theta}}\)
\(\displaystyle{y}={r}{\sin{\theta}}\)
b) From rectangular to polar:
\(\displaystyle{r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\)
\(\displaystyle{\cos{\theta}}={\frac{{{x}}}{{{r}}}},\ {\sin{\theta}}={\frac{{{y}}}{{{r}}}},\ {\tan{\theta}}={\frac{{{x}}}{{{y}}}}\)
Calcculation:
Converting into equivalent polar equation:
\(\displaystyle{y}=\ -{3}\)
Put \(\displaystyle{x}={r}{\cos{\theta}},\ {y}={r}{\sin{\theta}},\)
\(\displaystyle\Rightarrow\ {r}{\sin{\theta}}=\ -{3}\)
\(\displaystyle\Rightarrow\ {r}=\ -{\frac{{{3}}}{{{\sin{\theta}}}}}\)
Hence, desired equivalent polar equation would be \(\displaystyle{r}=\ -{\frac{{{3}}}{{{\sin{\theta}}}}}\)