Question

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

$\left(0,2\right)$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(0,2\right)$

Here$x=0\mathrm{and}y=2$.

Step 2. Find the value of r.

To find the value of r, use the formula $r=\sqrt{x^2+y^2}$:

$$\begin{gathered} r=\sqrt{{\left(0\right)}^2+{\left(2\right)}^2} \\ r=\sqrt{0+4} \\ r=\pm 2 \end{gathered}$$

Step 3. Find the value of θ.

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

$$\begin{gathered} \tan \theta =\frac{y}{x}\mathrm{then}\theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta ={\tan }^{-1}\left(\frac{2}{0}\right) \\ \theta ={\tan }^{-1}\left(\infty \right) \end{gathered}$$

$\theta =-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\left[\mathrm{Since}\tan \frac{\mathrm{\pi }}{2}=\infty ;\tan \frac{3\mathrm{\pi }}{2}=\infty \right]$

Step 4. Find the polar coordinates.

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

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

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

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

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

for any integer k.