Precalculus Examples

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

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.

$r = \sqrt{{(- \sqrt{6})}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

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 $- \sqrt{6}$ .

$r = \sqrt{{(- 1)}^{2} {\sqrt{6}}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

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

$r = \sqrt{1 {\sqrt{6}}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.3

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$r = \sqrt{{\sqrt{6}}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{{\sqrt{6}}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$r = \sqrt{{(6^{\frac{1}{2}})}^{2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

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

$r = \sqrt{6^{\frac{1}{2} \cdot 2} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.3

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

$r = \sqrt{6^{\frac{2}{2}} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$r = \sqrt{6^{\frac{2}{2}} + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4.2

Rewrite the expression.

$r = \sqrt{6 + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{6 + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.5

Evaluate the exponent.

$r = \sqrt{6 + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{6 + {(\sqrt{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

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

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Step 3.3.1

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

$r = \sqrt{6 + {(2^{\frac{1}{2}})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

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

$r = \sqrt{6 + 2^{\frac{1}{2} \cdot 2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.3

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

$r = \sqrt{6 + 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

$r = \sqrt{6 + 2^{\frac{2}{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.4.2

Rewrite the expression.

$r = \sqrt{6 + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{6 + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.5

Evaluate the exponent.

$r = \sqrt{6 + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{6 + 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Add $6$ and $2$ .

$r = \sqrt{8}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.5.1

Factor $4$ out of $8$ .

$r = \sqrt{4 (2)}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5.2

Rewrite $4$ as $2^{2}$ .

$r = \sqrt{2^{2} \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{2^{2} \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Pull terms out from under the radical.

$r = 2 \sqrt{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 2 \sqrt{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = 2 \sqrt{2}$

$θ = t a n^{- 1} (\frac{\sqrt{2}}{- \sqrt{6}})$

Step 5

The inverse tangent of $- \frac{\sqrt{3}}{3}$ is $θ = 150 °$ .

$r = 2 \sqrt{2}$

$θ = 150 °$

Step 6

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

$(2 \sqrt{2}, 150 °)$