Trigonometry Examples

$(2, \frac{7 π}{4})$

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.

$r = \sqrt{{(2)}^{2} + {(\frac{7 π}{4})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

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

$r = \sqrt{4 + {(\frac{7 π}{4})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

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 $\frac{7 π}{4}$ .

$r = \sqrt{4 + \frac{{(7 π)}^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Apply the product rule to $7 π$ .

$r = \sqrt{4 + \frac{7^{2} π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{4 + \frac{7^{2} π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

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

$r = \sqrt{4 + \frac{49 π^{2}}{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = \sqrt{4 + \frac{49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

To write $4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$r = \sqrt{4 \cdot \frac{16}{16} + \frac{49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Combine $4$ and $\frac{16}{16}$ .

$r = \sqrt{\frac{4 \cdot 16}{16} + \frac{49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Simplify the expression.

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Step 3.7.1

Combine the numerators over the common denominator.

$r = \sqrt{\frac{4 \cdot 16 + 49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7.2

Multiply $4$ by $16$ .

$r = \sqrt{\frac{64 + 49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{64 + 49 π^{2}}{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Rewrite $\sqrt{\frac{64 + 49 π^{2}}{16}}$ as $\frac{\sqrt{64 + 49 π^{2}}}{\sqrt{16}}$ .

$r = \frac{\sqrt{64 + 49 π^{2}}}{\sqrt{16}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Simplify the denominator.

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Step 3.9.1

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

$r = \frac{\sqrt{64 + 49 π^{2}}}{\sqrt{4^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.2

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

$r = \frac{\sqrt{64 + 49 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{64 + 49 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \frac{\sqrt{64 + 49 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = \frac{\sqrt{64 + 49 π^{2}}}{4}$

$θ = t a n^{- 1} (\frac{\frac{7 π}{4}}{2})$

Step 5

The inverse tangent of $\frac{7 π}{8}$ is $θ = 70.0094872 °$ .

$r = \frac{\sqrt{64 + 49 π^{2}}}{4}$

$θ = 70.0094872 °$

Step 6

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

$(\frac{\sqrt{64 + 49 π^{2}}}{4}, 70.0094872 °)$