Question

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

$\left(3,-3\right)$

Short answer

The polar coordinates are$\left(3\sqrt{2},-\frac{\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)\mathrm{and}\left(-3\sqrt{2},\frac{3\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)$, for any integer k.

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(3,-3\right)$

Here $x=3\mathrm{and}y=-3$.

Step 2. Find the value of r.

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

$$\begin{gathered} r=\sqrt{{\left(3\right)}^2+{\left(-3\right)}^2} \\ r=\sqrt{9+9} \\ r=\sqrt{18} \\ r=\pm 3\sqrt{2} \end{gathered}$$

Step 3. Find the value of θ.

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

$$\begin{gathered} \tan \theta =\frac{y}{x};\mathrm{then}\mathrm{\theta }={\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) \\ \mathrm{\theta }={\tan }^{-1}\left(\frac{-3}{3}\right) \\ \mathrm{\theta }={\tan }^{-1}\left(-1\right) \\ \mathrm{\theta }=-\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4} \end{gathered}$$

Step 4. Find the polar coordinates.

Take $r=3\sqrt{2},\theta =-\frac{\mathrm{\pi }}{4};\mathrm{then}\left(r,\theta \right)=\left(3\sqrt{2},-\frac{\mathrm{\pi }}{4}\right)$,

The coordinates of $\left(r,\theta \right)\mathrm{are}\left(3\sqrt{2},-\frac{\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)$, for any integer k.

Now take $\mathrm{r}=-3\sqrt{2},\mathrm{\theta }=\frac{3\mathrm{\pi }}{4};\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(-3\sqrt{2},\frac{3\mathrm{\pi }}{4}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(-3\sqrt{2},\frac{3\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(3\sqrt{2},-\frac{\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)\mathrm{and}\left(-3\sqrt{2},\frac{3\mathrm{\pi }}{4}+2\mathrm{k\pi }\right)$, for any integer k.