Trigonometry Examples

$(- 4, \frac{5 π}{4})$

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.

$r = \sqrt{{(- 4)}^{2} + {(\frac{5 π}{4})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

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

$r = \sqrt{16 + {(\frac{5 π}{4})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1

Apply the product rule to $\frac{5 π}{4}$ .

$r = \sqrt{16 + \frac{{(5 π)}^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Apply the product rule to $5 π$ .

$r = \sqrt{16 + \frac{5^{2} π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{16 + \frac{5^{2} π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

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

$r = \sqrt{16 + \frac{25 π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = \sqrt{16 + \frac{25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

To write $16$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$r = \sqrt{16 \cdot \frac{16}{16} + \frac{25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Combine $16$ and $\frac{16}{16}$ .

$r = \sqrt{\frac{16 \cdot 16}{16} + \frac{25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Simplify the expression.

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Step 3.7.1

Combine the numerators over the common denominator.

$r = \sqrt{\frac{16 \cdot 16 + 25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7.2

Multiply $16$ by $16$ .

$r = \sqrt{\frac{256 + 25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{256 + 25 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Rewrite $\sqrt{\frac{256 + 25 π^{2}}{16}}$ as $\frac{\sqrt{256 + 25 π^{2}}}{\sqrt{16}}$ .

$r = \frac{\sqrt{256 + 25 π^{2}}}{\sqrt{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Simplify the denominator.

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Step 3.9.1

Rewrite $16$ as $4^{2}$ .

$r = \frac{\sqrt{256 + 25 π^{2}}}{\sqrt{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.2

Pull terms out from under the radical, assuming positive real numbers.

$r = \frac{\sqrt{256 + 25 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{256 + 25 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{256 + 25 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \frac{\sqrt{256 + 25 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{\frac{5 π}{4}}{- 4})$

Step 5

The inverse tangent of $- \frac{5 π}{16}$ is $θ = 135.52769075 °$ .

$r = \frac{\sqrt{256 + 25 π^{2}}}{4}$

$θ = 135.52769075 °$

Step 6

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

$(\frac{\sqrt{256 + 25 π^{2}}}{4}, 135.52769075 °)$