Precalculus Examples

$(- \sqrt{3}, \frac{2 π}{3})$

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.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

Simplify the expression.

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Step 3.1.1

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.3

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.3

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4.2

Rewrite the expression.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.5

Evaluate the exponent.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

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

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Step 3.3.1

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

Apply the product rule to $2 π$ .

$r = \sqrt{3 + \frac{2^{2} π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{3 + \frac{2^{2} π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = \sqrt{3 + \frac{4 π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

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

$r = \sqrt{3 + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$r = \sqrt{3 \cdot \frac{9}{9} + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Combine $3$ and $\frac{9}{9}$ .

$r = \sqrt{\frac{3 \cdot 9}{9} + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Simplify the expression.

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Step 3.8.1

Combine the numerators over the common denominator.

$r = \sqrt{\frac{3 \cdot 9 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8.2

Multiply $3$ by $9$ .

$r = \sqrt{\frac{27 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{27 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Rewrite $\sqrt{\frac{27 + 4 π^{2}}{9}}$ as $\frac{\sqrt{27 + 4 π^{2}}}{\sqrt{9}}$ .

$r = \frac{\sqrt{27 + 4 π^{2}}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Simplify the denominator.

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Step 3.10.1

Rewrite $9$ as $3^{2}$ .

$r = \frac{\sqrt{27 + 4 π^{2}}}{\sqrt{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10.2

Pull terms out from under the radical, assuming positive real numbers.

$r = \frac{\sqrt{27 + 4 π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{27 + 4 π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{27 + 4 π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \frac{\sqrt{27 + 4 π^{2}}}{3}$

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

Step 5

The inverse tangent of $- \frac{2 π \sqrt{3}}{9}$ is $θ = 129.59052175 °$ .

$r = \frac{\sqrt{27 + 4 π^{2}}}{3}$

$θ = 129.59052175 °$

Step 6

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

$(\frac{\sqrt{27 + 4 π^{2}}}{3}, 129.59052175 °)$