Question

The equivalent polar coordinates for the given rectangular coordinates.
A rectangular coordinate is given as (0, -3).

Answer 1

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:
\(\displaystyle{r}=\pm\sqrt{{{x}^{2}+{y}^{2}}}\)
\(\displaystyle \cos{\theta}=\frac{x}{{r}}; \sin{\theta}=\frac{y}{{r}}; \tan{\theta}=\frac{x}{{y}}\)
Here \(\displaystyle{x}={0},{y}=-{3}\)
Converting into equivalent polar coordinates:
\(\displaystyle{r}=\pm\sqrt{{{x}^{2}+{y}^{2}}}\)
\(\displaystyle\Rightarrow{r}=\pm\sqrt{{{0}^{2}+{\left(-{3}\right)}^{2}}}\)
\(\displaystyle\Rightarrow{r}=\pm{3}\)
\(\displaystyle \cos{\theta}=\frac{x}{{r}}=\frac{0}{{3}}={0}\) {Taking positive value of r}
\(\displaystyle\Rightarrow\theta={90}^{\circ},{270}^{\circ}\)
\(\displaystyle \sin{\theta}=\frac{y}{{r}}=\frac{{-{3}}}{{3}}={1}\)
\(\displaystyle\Rightarrow\theta={270}^{\circ}\)
Hence, desired equivalent polar coordinates would be \(\displaystyle{\left({3},{270}^{\circ}\right)}\)