Convert to Polar Coordinates (5,135)

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Trigonometry

$(5, 135)$

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

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$r = \sqrt{{(5)}^{2} + {(135)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Find the magnitude of the polar coordinate.

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Raise $5$ to the power of $2$ .

$r = \sqrt{25 + {(135)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Raise $135$ to the power of $2$ .

$r = \sqrt{25 + 18225}$

$θ = t a n^{- 1} (\frac{y}{x})$

Add $25$ and $18225$ .

$r = \sqrt{18250}$

$θ = t a n^{- 1} (\frac{y}{x})$

Rewrite $18250$ as $5^{2} \cdot 730$ .

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Factor $25$ out of $18250$ .

$r = \sqrt{25 (730)}$

$θ = t a n^{- 1} (\frac{y}{x})$

Rewrite $25$ as $5^{2}$ .

$r = \sqrt{5^{2} \cdot 730}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{5^{2} \cdot 730}$

$θ = t a n^{- 1} (\frac{y}{x})$

Pull terms out from under the radical.

$r = 5 \sqrt{730}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 5 \sqrt{730}$

$θ = t a n^{- 1} (\frac{y}{x})$

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

$r = 5 \sqrt{730}$

$θ = t a n^{- 1} (\frac{135}{5})$

The inverse tangent of $27$ is $θ = 87.8789036 °$ .

$r = 5 \sqrt{730}$

$θ = 87.8789036 °$

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

$(5 \sqrt{730}, 87.8789036 °)$

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