Question

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(0,-3\right)$

Short answer

The polar coordinates are $\left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)\mathrm{and}\left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(0,-3\right)$

Here$x=0\mathrm{and}y=-3$

Step 2. Find the value of r.

To find the value of r, use the formula $r^2=x^2+y^2$,

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(0\right)}^2+{\left(-3\right)}^2} \\ r=\sqrt{0+9} \\ r=\pm 3 \end{gathered}$$

Step 3. Find the value of θ.

Use the formula $\tan \theta =\frac{y}{x}$,

$$\begin{gathered} \tan \theta =\frac{y}{x} \\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta ={\tan }^{-1}\left(\frac{-3}{0}\right) \\ \theta ={\tan }^{-1}\left(\infty \right) \end{gathered}$$

role="math" localid="1649230458419" $\theta =-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\left[\mathrm{since}\tan \frac{\mathrm{\pi }}{2}=\infty ;\tan \frac{3\mathrm{\pi }}{2}=\infty \right]$

Step 4. Find the polar coordinates.

Take $r=-3,\theta =-\frac{\mathrm{\pi }}{2},\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(-3,-\frac{\mathrm{\pi }}{2}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$for any integer k.

Now take $\mathrm{r}=3,\mathrm{\theta }=\frac{\mathrm{\pi }}{2},\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(3,\frac{\mathrm{\pi }}{2}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)\mathrm{and}\left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$,

for any integer k.