Precalculus Examples

$(- \frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{3})$

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.

$r = \sqrt{{(- \frac{3 \sqrt{2}}{2})}^{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

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 $- \frac{3 \sqrt{2}}{2}$ .

$r = \sqrt{{(- 1)}^{2} {(\frac{3 \sqrt{2}}{2})}^{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{{(3 \sqrt{2})}^{2}}{2^{2}}) + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.3

Apply the product rule to $3 \sqrt{2}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}) + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{{(- 1)}^{2} (\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}) + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Simplify the expression.

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Step 3.2.1

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

$r = \sqrt{1 (\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}) + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Multiply $\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

$r = \sqrt{\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{3^{2} {\sqrt{2}}^{2}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Raise $3$ to the power of $2$ .

$r = \sqrt{\frac{9 {\sqrt{2}}^{2}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

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

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Step 3.3.2.1

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

$r = \sqrt{\frac{9 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2.2

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

$r = \sqrt{\frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2.3

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

$r = \sqrt{\frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.4.1

Cancel the common factor.

$r = \sqrt{\frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2.4.2

Rewrite the expression.

$r = \sqrt{\frac{9 \cdot 2}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9 \cdot 2}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2.5

Evaluate the exponent.

$r = \sqrt{\frac{9 \cdot 2}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9 \cdot 2}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9 \cdot 2}{2^{2}} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Simplify terms.

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Step 3.4.1

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

$r = \sqrt{\frac{9 \cdot 2}{4} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.2

Multiply $9$ by $2$ .

$r = \sqrt{\frac{18}{4} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.3

Cancel the common factor of $18$ and $4$ .

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Step 3.4.3.1

Factor $2$ out of $18$ .

$r = \sqrt{\frac{2 (9)}{4} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Factor $2$ out of $4$ .

$r = \sqrt{\frac{2 \cdot 9}{2 \cdot 2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.3.2.2

Cancel the common factor.

$r = \sqrt{\frac{2 \cdot 9}{2 \cdot 2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.3.2.3

Rewrite the expression.

$r = \sqrt{\frac{9}{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.4

Cancel the common factor of $3$ .

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Step 3.4.4.1

Cancel the common factor.

$r = \sqrt{\frac{9}{2} + {(- \frac{3 \sqrt{2}}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.4.2

Divide $\sqrt{2}$ by $1$ .

$r = \sqrt{\frac{9}{2} + {(- \sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + {(- \sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

Apply the product rule to $- \sqrt{2}$ .

$r = \sqrt{\frac{9}{2} + {(- 1)}^{2} {\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.5.2

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

$r = \sqrt{\frac{9}{2} + 1 {\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.5.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$r = \sqrt{\frac{9}{2} + {\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + {\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6

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

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Step 3.4.6.1

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

$r = \sqrt{\frac{9}{2} + {(2^{\frac{1}{2}})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6.2

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

$r = \sqrt{\frac{9}{2} + 2^{\frac{1}{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6.3

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

$r = \sqrt{\frac{9}{2} + 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

$r = \sqrt{\frac{9}{2} + 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6.4.2

Rewrite the expression.

$r = \sqrt{\frac{9}{2} + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6.5

Evaluate the exponent.

$r = \sqrt{\frac{9}{2} + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{9}{2} + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r = \sqrt{\frac{9}{2} + 2 \cdot \frac{2}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

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

$r = \sqrt{\frac{9}{2} + \frac{2 \cdot 2}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Combine the numerators over the common denominator.

$r = \sqrt{\frac{9 + 2 \cdot 2}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $2$ by $2$ .

$r = \sqrt{\frac{9 + 4}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8.2

Add $9$ and $4$ .

$r = \sqrt{\frac{13}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{13}{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Rewrite $\sqrt{\frac{13}{2}}$ as $\frac{\sqrt{13}}{\sqrt{2}}$ .

$r = \frac{\sqrt{13}}{\sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Multiply $\frac{\sqrt{13}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{13}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11

Combine and simplify the denominator.

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Step 3.11.1

Multiply $\frac{\sqrt{13}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{13} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.2

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

$r = \frac{\sqrt{13} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.3

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

$r = \frac{\sqrt{13} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.4

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

$r = \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.5

Add $1$ and $1$ .

$r = \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6

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

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Step 3.11.6.1

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

$r = \frac{\sqrt{13} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6.2

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

$r = \frac{\sqrt{13} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6.3

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

$r = \frac{\sqrt{13} \sqrt{2}}{2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6.4

Cancel the common factor of $2$ .

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Step 3.11.6.4.1

Cancel the common factor.

$r = \frac{\sqrt{13} \sqrt{2}}{2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6.4.2

Rewrite the expression.

$r = \frac{\sqrt{13} \sqrt{2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{13} \sqrt{2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11.6.5

Evaluate the exponent.

$r = \frac{\sqrt{13} \sqrt{2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{13} \sqrt{2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{13} \sqrt{2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.12

Simplify the numerator.

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Step 3.12.1

Combine using the product rule for radicals.

$r = \frac{\sqrt{13 \cdot 2}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.12.2

Multiply $13$ by $2$ .

$r = \frac{\sqrt{26}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{26}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{26}}{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \frac{\sqrt{26}}{2}$

$θ = t a n^{- 1} (\frac{- \frac{3 \sqrt{2}}{3}}{- \frac{3 \sqrt{2}}{2}})$

Step 5

The inverse tangent of $\frac{2}{3}$ is $θ = 213.69006752 °$ .

$r = \frac{\sqrt{26}}{2}$

$θ = 213.69006752 °$

Step 6

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

$(\frac{\sqrt{26}}{2}, 213.69006752 °)$