Calculus Examples

$(7, 7)$

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.

$r = \sqrt{{(7)}^{2} + {(7)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Find the magnitude of the polar coordinate.

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Raise $7$ to the power of $2$ .

$r = \sqrt{49 + {(7)}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Raise $7$ to the power of $2$ .

$r = \sqrt{49 + 49}$

$θ = t a n^{- 1} (\frac{y}{x})$

Add $49$ and $49$ .

$r = \sqrt{98}$

$θ = t a n^{- 1} (\frac{y}{x})$

Rewrite $98$ as $7^{2} \cdot 2$ .

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Factor $49$ out of $98$ .

$r = \sqrt{49 (2)}$

$θ = t a n^{- 1} (\frac{y}{x})$

Rewrite $49$ as $7^{2}$ .

$r = \sqrt{7^{2} \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{7^{2} \cdot 2}$

$θ = t a n^{- 1} (\frac{y}{x})$

Pull terms out from under the radical.

$r = 7 \sqrt{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 7 \sqrt{2}$

$θ = t a n^{- 1} (\frac{y}{x})$

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

$r = 7 \sqrt{2}$

$θ = t a n^{- 1} (\frac{7}{7})$

The inverse tangent of $1$ is $θ = 45 °$ .

$r = 7 \sqrt{2}$

$θ = 45 °$

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

$(7 \sqrt{2}, 45 °)$