Trigonometry Examples

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

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.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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 $- \sqrt{3}$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.3

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

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 $3^{\frac{1}{2}}$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.3

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4.2

Rewrite the expression.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.5

Evaluate the exponent.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Simplify the expression.

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Step 3.3.1

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

$r = \sqrt{3 + 9}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

Add $3$ and $9$ .

$r = \sqrt{12}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{12}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.4.1

Factor $4$ out of $12$ .

$r = \sqrt{4 (3)}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.2

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

$r = \sqrt{2^{2} \cdot 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{2^{2} \cdot 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

Pull terms out from under the radical.

$r = 2 \sqrt{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 2 \sqrt{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = 2 \sqrt{3}$

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

Step 5

The inverse tangent of $- \sqrt{3}$ is $θ = 120 °$ .

$r = 2 \sqrt{3}$

$θ = 120 °$

Step 6

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

$(2 \sqrt{3}, 120 °)$