Question

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

$\left(1,0\right)$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(1,0\right)$

Here$x=1\mathrm{and}y=0$.

Step 2. Find the value of r.

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

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(1\right)}^2+{\left(0\right)}^2} \\ r=\sqrt{1+0} \\ r=\pm 1 \end{gathered}$$

Step 3. Find the value of θ.

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

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

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

Step 4. Find the polar coordinates.

Now take $r=-1,\theta =\mathrm{\pi }$,

Then $\left(\mathrm{r},\mathrm{\theta }\right)=\left(-1,\mathrm{\pi }\right)$

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

Now take $\mathrm{r}=1,\mathrm{\theta }=2\mathrm{\pi }$,

Then $\left(\mathrm{r},\mathrm{\theta }\right)=\left(1,2\mathrm{\pi }\right)$.

The coordinates of localid="1649229808751" $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(1,2\mathrm{k\pi }\right)$, for any integer k.

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

for any integer k.