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Trigonometry Examples

(- 2, 9)

$(- 2, 9)$

Step 1

Convert from rectangular coordinates

(x, y)

$(x, y)$ to polar coordinates

(r, θ)

$(r, θ)$ using the conversion formulas.

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

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

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 2

Replace

x

$x$ and

y

$y$ with the actual values.

r = \sqrt{{(- 2)}^{2} + {(9)}^{2}}

$r = \sqrt{{(- 2)}^{2} + {(9)}^{2}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

r = \sqrt{4 + {(9)}^{2}}

$r = \sqrt{4 + {(9)}^{2}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Raise

9

$9$ to the power of

2

$2$ .

r = \sqrt{4 + 81}

$r = \sqrt{4 + 81}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Add

4

$4$ and

81

$81$ .

r = \sqrt{85}

$r = \sqrt{85}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

r = \sqrt{85}

$r = \sqrt{85}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

Replace

x

$x$ and

y

$y$ with the actual values.

r = \sqrt{85}

$r = \sqrt{85}$

θ = t a n^{- 1} (\frac{9}{- 2})

$θ = t a n^{- 1} (\frac{9}{- 2})$

Step 5

The inverse tangent of

- \frac{9}{2}

$- \frac{9}{2}$ is

θ = 102.5288077 °

$θ = 102.5288077 °$ .

r = \sqrt{85}

$r = \sqrt{85}$

θ = 102.5288077 °

$θ = 102.5288077 °$

Step 6

This is the result of the conversion to polar coordinates in

(r, θ)

$(r, θ)$ form.

(\sqrt{85}, 102.5288077 °)

$(\sqrt{85}, 102.5288077 °)$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$