Trigonometry Examples

(0, - \frac{7 π}{6})

$(0, - \frac{7 π}{6})$

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{{(0)}^{2} + {(- \frac{7 π}{6})}^{2}}

$r = \sqrt{{(0)}^{2} + {(- \frac{7 π}{6})}^{2}}$

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

Raising

0

$0$ to any positive power yields

0

$0$ .

r = \sqrt{0 + {(- \frac{7 π}{6})}^{2}}

$r = \sqrt{0 + {(- \frac{7 π}{6})}^{2}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

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

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Step 3.2.1

Apply the product rule to

- \frac{7 π}{6}

$- \frac{7 π}{6}$ .

r = \sqrt{0 + {(- 1)}^{2} {(\frac{7 π}{6})}^{2}}

$r = \sqrt{0 + {(- 1)}^{2} {(\frac{7 π}{6})}^{2}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Apply the product rule to

\frac{7 π}{6}

$\frac{7 π}{6}$ .

r = \sqrt{0 + {(- 1)}^{2} (\frac{{(7 π)}^{2}}{6^{2}})}

$r = \sqrt{0 + {(- 1)}^{2} (\frac{{(7 π)}^{2}}{6^{2}})}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.3

Apply the product rule to

7 π

$7 π$ .

r = \sqrt{0 + {(- 1)}^{2} (\frac{7^{2} π^{2}}{6^{2}})}

$r = \sqrt{0 + {(- 1)}^{2} (\frac{7^{2} π^{2}}{6^{2}})}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

r = \sqrt{0 + {(- 1)}^{2} (\frac{7^{2} π^{2}}{6^{2}})}

$r = \sqrt{0 + {(- 1)}^{2} (\frac{7^{2} π^{2}}{6^{2}})}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

r = \sqrt{0 + 1 (\frac{7^{2} π^{2}}{6^{2}})}

$r = \sqrt{0 + 1 (\frac{7^{2} π^{2}}{6^{2}})}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Multiply

\frac{7^{2} π^{2}}{6^{2}}

$\frac{7^{2} π^{2}}{6^{2}}$ by

1

$1$ .

r = \sqrt{0 + \frac{7^{2} π^{2}}{6^{2}}}

$r = \sqrt{0 + \frac{7^{2} π^{2}}{6^{2}}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

Raise

7

$7$ to the power of

2

$2$ .

r = \sqrt{0 + \frac{49 π^{2}}{6^{2}}}

$r = \sqrt{0 + \frac{49 π^{2}}{6^{2}}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Raise

6

$6$ to the power of

2

$2$ .

r = \sqrt{0 + \frac{49 π^{2}}{36}}

$r = \sqrt{0 + \frac{49 π^{2}}{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

Add

0

$0$ and

\frac{49 π^{2}}{36}

$\frac{49 π^{2}}{36}$ .

r = \sqrt{\frac{49 π^{2}}{36}}

$r = \sqrt{\frac{49 π^{2}}{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Rewrite

\sqrt{\frac{49 π^{2}}{36}}

$\sqrt{\frac{49 π^{2}}{36}}$ as

\frac{\sqrt{49 π^{2}}}{\sqrt{36}}

$\frac{\sqrt{49 π^{2}}}{\sqrt{36}}$ .

r = \frac{\sqrt{49 π^{2}}}{\sqrt{36}}

$r = \frac{\sqrt{49 π^{2}}}{\sqrt{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Rewrite

49 π^{2}

$49 π^{2}$ as

{(7 π)}^{2}

${(7 π)}^{2}$ .

r = \frac{\sqrt{{(7 π)}^{2}}}{\sqrt{36}}

$r = \frac{\sqrt{{(7 π)}^{2}}}{\sqrt{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9.2

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

r = \frac{7 π}{\sqrt{36}}

$r = \frac{7 π}{\sqrt{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

r = \frac{7 π}{\sqrt{36}}

$r = \frac{7 π}{\sqrt{36}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Simplify the denominator.

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Step 3.10.1

Rewrite

36

$36$ as

6^{2}

$6^{2}$ .

r = \frac{7 π}{\sqrt{6^{2}}}

$r = \frac{7 π}{\sqrt{6^{2}}}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10.2

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

r = \frac{7 π}{6}

$r = \frac{7 π}{6}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

r = \frac{7 π}{6}

$r = \frac{7 π}{6}$

θ = t a n^{- 1} (\frac{y}{x})

$θ = t a n^{- 1} (\frac{y}{x})$

r = \frac{7 π}{6}

$r = \frac{7 π}{6}$

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 = \frac{7 π}{6}

$r = \frac{7 π}{6}$

θ = t a n^{- 1} (\frac{- \frac{7 π}{6}}{0})

$θ = t a n^{- 1} (\frac{- \frac{7 π}{6}}{0})$

Step 5

The inverse tangent of Undefined is

θ = 270 °

$θ = 270 °$ .

r = \frac{7 π}{6}

$r = \frac{7 π}{6}$

θ = 270 °

$θ = 270 °$

Step 6

This is the result of the conversion to polar coordinates in

(r, θ)

$(r, θ)$ form.

(\frac{7 π}{6}, 270 °)

$(\frac{7 π}{6}, 270 °)$