Precalculus Examples

$(- \frac{21}{8}, \frac{13}{24})$

Step 1

Convert from rectangular coordinates $(x, y)$ to polar coordinates $(r, θ)$ using the conversion formulas.

$r = \sqrt{x^{2} + y^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 2

Replace $x$ and $y$ with the actual values.

$r = \sqrt{{(- \frac{21}{8})}^{2} + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.1

Apply the product rule to $- \frac{21}{8}$ .

$r = \sqrt{{(- 1)}^{2} {(\frac{21}{8})}^{2} + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

Apply the product rule to $\frac{21}{8}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{21^{2}}{8^{2}}) + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{{(- 1)}^{2} (\frac{21^{2}}{8^{2}}) + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Raise $- 1$ to the power of $2$ .

$r = \sqrt{1 (\frac{21^{2}}{8^{2}}) + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Multiply $\frac{21^{2}}{8^{2}}$ by $1$ .

$r = \sqrt{\frac{21^{2}}{8^{2}} + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Raise $21$ to the power of $2$ .

$r = \sqrt{\frac{441}{8^{2}} + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

Raise $8$ to the power of $2$ .

$r = \sqrt{\frac{441}{64} + {(\frac{13}{24})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Apply the product rule to $\frac{13}{24}$ .

$r = \sqrt{\frac{441}{64} + \frac{13^{2}}{24^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Raise $13$ to the power of $2$ .

$r = \sqrt{\frac{441}{64} + \frac{169}{24^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Raise $24$ to the power of $2$ .

$r = \sqrt{\frac{441}{64} + \frac{169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

To write $\frac{441}{64}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$r = \sqrt{\frac{441}{64} \cdot \frac{9}{9} + \frac{169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Write each expression with a common denominator of $576$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.10.1

Multiply $\frac{441}{64}$ by $\frac{9}{9}$ .

$r = \sqrt{\frac{441 \cdot 9}{64 \cdot 9} + \frac{169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10.2

Multiply $64$ by $9$ .

$r = \sqrt{\frac{441 \cdot 9}{576} + \frac{169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{441 \cdot 9}{576} + \frac{169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11

Combine the numerators over the common denominator.

$r = \sqrt{\frac{441 \cdot 9 + 169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $441$ by $9$ .

$r = \sqrt{\frac{3969 + 169}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.12.2

Add $3969$ and $169$ .

$r = \sqrt{\frac{4138}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{4138}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.13

Cancel the common factor of $4138$ and $576$ .

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Step 3.13.1

Factor $2$ out of $4138$ .

$r = \sqrt{\frac{2 (2069)}{576}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.13.2

Cancel the common factors.

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Step 3.13.2.1

Factor $2$ out of $576$ .

$r = \sqrt{\frac{2 \cdot 2069}{2 \cdot 288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.13.2.2

Cancel the common factor.

$r = \sqrt{\frac{2 \cdot 2069}{2 \cdot 288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.13.2.3

Rewrite the expression.

$r = \sqrt{\frac{2069}{288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{2069}{288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{2069}{288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.14

Rewrite $\sqrt{\frac{2069}{288}}$ as $\frac{\sqrt{2069}}{\sqrt{288}}$ .

$r = \frac{\sqrt{2069}}{\sqrt{288}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.15

Simplify the denominator.

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Step 3.15.1

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 3.15.1.1

Factor $144$ out of $288$ .

$r = \frac{\sqrt{2069}}{\sqrt{144 (2)}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.15.1.2

Rewrite $144$ as $12^{2}$ .

$r = \frac{\sqrt{2069}}{\sqrt{12^{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2069}}{\sqrt{12^{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.15.2

Pull terms out from under the radical.

$r = \frac{\sqrt{2069}}{12 \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2069}}{12 \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.16

Multiply $\frac{\sqrt{2069}}{12 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{2069}}{12 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17

Combine and simplify the denominator.

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Step 3.17.1

Multiply $\frac{\sqrt{2069}}{12 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \sqrt{2} \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.2

Move $\sqrt{2}$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 (\sqrt{2} \sqrt{2})}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.3

Raise $\sqrt{2}$ to the power of $1$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 (\sqrt{2} \sqrt{2})}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.4

Raise $\sqrt{2}$ to the power of $1$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 (\sqrt{2} \sqrt{2})}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{\sqrt{2069} \sqrt{2}}{12 {\sqrt{2}}^{1 + 1}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.6

Add $1$ and $1$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 {\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.17.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 {(2^{\frac{1}{2}})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2^{\frac{1}{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7.3

Combine $\frac{1}{2}$ and $2$ .

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7.4

Cancel the common factor of $2$ .

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Step 3.17.7.4.1

Cancel the common factor.

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7.4.2

Rewrite the expression.

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.17.7.5

Evaluate the exponent.

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2069} \sqrt{2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.18

Simplify the numerator.

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Step 3.18.1

Combine using the product rule for radicals.

$r = \frac{\sqrt{2069 \cdot 2}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.18.2

Multiply $2069$ by $2$ .

$r = \frac{\sqrt{4138}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{4138}}{12 \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.19

Multiply $12$ by $2$ .

$r = \frac{\sqrt{4138}}{24}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{4138}}{24}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

Replace $x$ and $y$ with the actual values.

$r = \frac{\sqrt{4138}}{24}$

$θ = t a n^{- 1} (\frac{\frac{13}{24}}{- \frac{21}{8}})$

Step 5

The inverse tangent of $- \frac{13}{63}$ is $θ = 168.34070734 °$ .

$r = \frac{\sqrt{4138}}{24}$

$θ = 168.34070734 °$

Step 6

This is the result of the conversion to polar coordinates in $(r, θ)$ form.

$(\frac{\sqrt{4138}}{24}, 168.34070734 °)$