Question

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

$\left(0,0\right)$

Short answer

The polar coordinates are $\left(0,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right)\mathrm{and}\left(0,2\mathrm{k\pi }\right)$, for any integerk.

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(0,0\right)$

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

Step 2. Find the value of r.

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

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

Step 3. Find the value of θ.

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

localid="1649227271585" $$\begin{gathered} \tan \theta =\frac{y}{x} \\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta ={\tan }^{-1}\left(\frac{0}{0}\right) \\ \theta ={\tan }^{-1}\left(0\right) \end{gathered}$$

$\theta =0,\mathrm{\pi },2\mathrm{\pi }$ localid="1649227391599" $\left(\mathrm{Since}\tan 2\mathrm{\pi }=0;\tan \mathrm{\pi }=0\right)$)

Step 4. Find the polar coordinates.

Now take $r=0,\theta =\mathrm{\pi }$,

Then $\left(\mathrm{r},\mathrm{\theta }\right)=\left(0,\mathrm{\pi }\right)$

The coordinates will become localid="1649227976138" $\left(0,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right)\mathrm{and}\left(0,2\mathrm{k\pi }\right)$ for any integer k.