Q 24. - Calculus

Question

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

$\left(0,0\right)$

Step-by-Step Solution

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 integer k.

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 $\tan \theta =\frac{y}{x}$,

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

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

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 $\left(0,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right) \mathrm{and} \left(0,2\mathrm{k\pi }\right)$ for any integer k.