Precalculus Examples

$(4, 1)$

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.

$r = \sqrt{{(4)}^{2} + {(1)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

Raise $4$ to the power of $2$ .

$r = \sqrt{16 + {(1)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

One to any power is one.

$r = \sqrt{16 + 1}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Add $16$ and $1$ .

$r = \sqrt{17}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{17}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \sqrt{17}$

$θ = t a n^{- 1} (\frac{1}{4})$

Step 5

The inverse tangent of $\frac{1}{4}$ is $θ = 14.03624346 °$ .

$r = \sqrt{17}$

$θ = 14.03624346 °$

Step 6

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

$(\sqrt{17}, 14.03624346 °)$