Q 31. - Calculus

Question

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

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

Step-by-Step Solution

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 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.