Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(6, 2 \sqrt{3})$

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given rectangular coordinate is:

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

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

Step 2. Find the value of r.

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

$r = \sqrt{{(6)}^{2} + {(2 \sqrt{3})}^{2}} r = \sqrt{36 + 12} r = \sqrt{48} r = \pm 4 \sqrt{3}$

Step 3. Find the value of θ.

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

$\tan θ = \frac{y}{x}; then θ = \tan^{- 1} (\frac{y}{x}) θ = \tan^{- 1} (\frac{2 \sqrt{3}}{6}) θ = \tan^{- 1} (\frac{\sqrt{3}}{3}); θ = \tan^{- 1} (\frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}) θ = \tan^{- 1} (\frac{1}{\sqrt{3}}) θ = \frac{π}{6}, \frac{7 π}{6}$

Step 4. Find the polar coordinates.

Take $r = 4 \sqrt{3}, θ = \frac{π}{6}; then (r, θ) = (4 \sqrt{3}, \frac{π}{6})$ ,

The coordinates of $(r, θ) are (4 \sqrt{3}, \frac{π}{6} + 2 kπ)$ , for any integer k.

Now take $r = - 4 \sqrt{3}, θ = \frac{7 π}{6}; then (r, θ) = (- 4 \sqrt{3}, \frac{7 π}{6})$ ,

The coordinates of $(r, θ) are (- 4 \sqrt{3}, \frac{7 π}{6} + 2 kπ)$ , for any integer k.

Therefore, all the polar coordinates are $(4 \sqrt{3}, \frac{π}{6} + 2 kπ) and (- 4 \sqrt{3}, \frac{7 π}{6} + 2 kπ)$ , for any integer k.

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Calculus

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Short Answer

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(6, 2 \sqrt{3})$

Step by Step Solution

Step 1. Given information.

Step 2. Find the value of r.

Step 3. Find the value of θ.

Step 4. Find the polar coordinates.

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In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.6,23

Step by Step Solution

Step 1. Given information.

Step 2. Find the value of r.

Step 3. Find the value of θ.

Step 4. Find the polar coordinates.

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In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(6, 2 \sqrt{3})$