Precalculus Examples

$(1, 3)$

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.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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One to any power is one.

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

$θ = t a n^{- 1} (\frac{y}{x})$

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

$r = \sqrt{1 + 9}$

$θ = t a n^{- 1} (\frac{y}{x})$

Add $1$ and $9$ .

$r = \sqrt{10}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{10}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \sqrt{10}$

$θ = t a n^{- 1} (\frac{3}{1})$

Step 5

The inverse tangent of $3$ is $θ = 71.56505117 °$ .

$r = \sqrt{10}$

$θ = 71.56505117 °$

Step 6

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

$(\sqrt{10}, 71.56505117 °)$

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