Trigonometry Examples

$(1, \sqrt{3})$

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.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3

Find the magnitude of the polar coordinate.

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Step 3.1

One to any power is one.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

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

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.3

Combine $\frac{1}{2}$ and $2$ .

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

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

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.4.2

Rewrite the expression.

$r = \sqrt{1 + 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{1 + 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2.5

Evaluate the exponent.

$r = \sqrt{1 + 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{1 + 3}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Simplify the expression.

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Step 3.3.1

Add $1$ and $3$ .

$r = \sqrt{4}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3.2

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

$r = \sqrt{2^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{2^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = 2$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 2$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = 2$

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

Step 5

The inverse tangent of $\sqrt{3}$ is $θ = 60 °$ .

$r = 2$

$θ = 60 °$

Step 6

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

$(2, 60 °)$