Trigonometry Examples

$(4, \frac{2 π}{3})$

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.

$r = \sqrt{{(4)}^{2} + {(\frac{2 π}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

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

$r = \sqrt{16 + {(\frac{2 π}{3})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

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 $\frac{2 π}{3}$ .

$r = \sqrt{16 + \frac{{(2 π)}^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Apply the product rule to $2 π$ .

$r = \sqrt{16 + \frac{2^{2} π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{16 + \frac{2^{2} π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

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

$r = \sqrt{16 + \frac{4 π^{2}}{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = \sqrt{16 + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

To write $16$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$r = \sqrt{16 \cdot \frac{9}{9} + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Combine $16$ and $\frac{9}{9}$ .

$r = \sqrt{\frac{16 \cdot 9}{9} + \frac{4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Simplify the expression.

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Step 3.7.1

Combine the numerators over the common denominator.

$r = \sqrt{\frac{16 \cdot 9 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7.2

Multiply $16$ by $9$ .

$r = \sqrt{\frac{144 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{144 + 4 π^{2}}{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Rewrite $\sqrt{\frac{144 + 4 π^{2}}{9}}$ as $\frac{\sqrt{144 + 4 π^{2}}}{\sqrt{9}}$ .

$r = \frac{\sqrt{144 + 4 π^{2}}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Rewrite $144 + 4 π^{2}$ as $2^{2} (36 + π^{2})$ .

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Step 3.9.1.1

Factor $4$ out of $144$ .

$r = \frac{\sqrt{4 (36) + 4 π^{2}}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.1.2

Factor $4$ out of $4 π^{2}$ .

$r = \frac{\sqrt{4 (36) + 4 (π^{2})}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.1.3

Factor $4$ out of $4 (36) + 4 (π^{2})$ .

$r = \frac{\sqrt{4 (36 + π^{2})}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.1.4

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

$r = \frac{\sqrt{2^{2} (36 + π^{2})}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{2^{2} (36 + π^{2})}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.2

Pull terms out from under the radical.

$r = \frac{2 \sqrt{36 + π^{2}}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{2 \sqrt{36 + π^{2}}}{\sqrt{9}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Simplify the denominator.

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Step 3.10.1

Rewrite $9$ as $3^{2}$ .

$r = \frac{2 \sqrt{36 + π^{2}}}{\sqrt{3^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10.2

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

$r = \frac{2 \sqrt{36 + π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{2 \sqrt{36 + π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{2 \sqrt{36 + π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \frac{2 \sqrt{36 + π^{2}}}{3}$

$θ = t a n^{- 1} (\frac{\frac{2 π}{3}}{4})$

Step 5

The inverse tangent of $\frac{π}{6}$ is $θ = 27.63649933 °$ .

$r = \frac{2 \sqrt{36 + π^{2}}}{3}$

$θ = 27.63649933 °$

Step 6

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

$(\frac{2 \sqrt{36 + π^{2}}}{3}, 27.63649933 °)$