Question

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

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

2020-11-02

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 $$\displaystyle{\left({3},{270}^{\circ}\right)}$$