Trigonometry Examples

$(- \frac{\sqrt{3}}{2}, \frac{1}{2})$

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.

$r = \sqrt{{(- \frac{\sqrt{3}}{2})}^{2} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

Use the power rule

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

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Step 3.1.1

Apply the product rule to

$- \frac{\sqrt{3}}{2}$ .

$r = \sqrt{{(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

Apply the product rule to

$\frac{\sqrt{3}}{2}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{{\sqrt{3}}^{2}}{2^{2}}) + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{{(- 1)}^{2} (\frac{{\sqrt{3}}^{2}}{2^{2}}) + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Simplify the expression.

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Step 3.2.1

Raise

$- 1$ to the power of

$2$ .

$r = \sqrt{1 (\frac{{\sqrt{3}}^{2}}{2^{2}}) + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Multiply

$\frac{{\sqrt{3}}^{2}}{2^{2}}$ by

$1$ .

$r = \sqrt{\frac{{\sqrt{3}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{{\sqrt{3}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 3.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$r = \sqrt{\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$r = \sqrt{\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.3

Combine

$\frac{1}{2}$ and

$2$ .

$r = \sqrt{\frac{3^{\frac{2}{2}}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.4

Cancel the common factor of

$2$ .

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Step 3.3.4.1

Cancel the common factor.

$r = \sqrt{\frac{3^{\frac{2}{2}}}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.4.2

Rewrite the expression.

$r = \sqrt{\frac{3}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{3}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.5

Evaluate the exponent.

$r = \sqrt{\frac{3}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{\frac{3}{2^{2}} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

Simplify the expression.

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Step 3.4.1

Raise

$2$ to the power of

$2$ .

$r = \sqrt{\frac{3}{4} + {(\frac{1}{2})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.2

Apply the product rule to

$\frac{1}{2}$ .

$r = \sqrt{\frac{3}{4} + \frac{1^{2}}{2^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.3

One to any power is one.

$r = \sqrt{\frac{3}{4} + \frac{1}{2^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.4

Raise

$2$ to the power of

$2$ .

$r = \sqrt{\frac{3}{4} + \frac{1}{4}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.5

Combine the numerators over the common denominator.

$r = \sqrt{\frac{3 + 1}{4}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.6

Add

$3$ and

$1$ .

$r = \sqrt{\frac{4}{4}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.7

Divide

$4$ by

$4$ .

$r = \sqrt{1}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4.8

Any root of

$1$ is

$1$ .

$r = 1$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 1$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 1$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

Replace

$x$ and

$y$ with the actual values.

$r = 1$

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

Step 5

The inverse tangent of

$- \frac{\sqrt{3}}{3}$ is

$θ = 150 °$ .

$r = 1$

$θ = 150 °$

Step 6

This is the result of the conversion to polar coordinates in

$(r, θ)$ form.

$(1, 150 °)$