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Expert-verified Found in: Page 730 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.$\left(6,2\sqrt{3}\right)$

The polar coordinates are $\left(4\sqrt{3},\frac{\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)\mathrm{and}\left(-4\sqrt{3},\frac{7\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)$, for any integer k.

See the step by step solution

## Step 1. Given information.

The given rectangular coordinate is:

$\left(6,2\sqrt{3}\right)$

Here $x=6\mathrm{and}y=2\sqrt{3}$.

## Step 2. Find the value of r.

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

$r=\sqrt{{\left(6\right)}^{2}+{\left(2\sqrt{3}\right)}^{2}}\phantom{\rule{0ex}{0ex}}r=\sqrt{36+12}\phantom{\rule{0ex}{0ex}}r=\sqrt{48}\phantom{\rule{0ex}{0ex}}r=±4\sqrt{3}$

## Step 3. Find the value of θ.

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

$\mathrm{tan}\theta =\frac{y}{x};\mathrm{then}\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{2\sqrt{3}}{6}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{\sqrt{3}}{3}\right);\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{\sqrt{3}}{\sqrt{3}·\sqrt{3}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{1}{\sqrt{3}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\theta }=\frac{\mathrm{\pi }}{6},\frac{7\mathrm{\pi }}{6}$

## Step 4. Find the polar coordinates.

Take $r=4\sqrt{3},\mathrm{\theta }=\frac{\mathrm{\pi }}{6};\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(4\sqrt{3},\frac{\mathrm{\pi }}{6}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(4\sqrt{3},\frac{\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)$, for any integer k.

Now take $\mathrm{r}=-4\sqrt{3},\mathrm{\theta }=\frac{7\mathrm{\pi }}{6};\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(-4\sqrt{3},\frac{7\mathrm{\pi }}{6}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(-4\sqrt{3},\frac{7\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(4\sqrt{3},\frac{\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)\mathrm{and}\left(-4\sqrt{3},\frac{7\mathrm{\pi }}{6}+2\mathrm{k\pi }\right)$, for any integer k. ### Want to see more solutions like these? 