Question

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

$\left(-2,0\right)$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(-2,0\right)$

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

Step 2. Find the value of r.

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

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

Step 3. Find the value of θ.

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

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

Step 4. Find the polar coordinates.

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

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

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

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(-2,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(2,2\mathrm{k\pi }\right)\mathrm{and}\left(-2,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right)$, for any integer k.