Trigonometry Examples

$(- \frac{3}{5}, \frac{4}{5})$

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.

$r = \sqrt{{(- \frac{3}{5})}^{2} + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

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 $- \frac{3}{5}$ .

$r = \sqrt{{(- 1)}^{2} {(\frac{3}{5})}^{2} + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.1.2

Apply the product rule to $\frac{3}{5}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{3^{2}}{5^{2}}) + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = \sqrt{{(- 1)}^{2} (\frac{3^{2}}{5^{2}}) + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.2

Raise $- 1$ to the power of $2$ .

$r = \sqrt{1 (\frac{3^{2}}{5^{2}}) + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.3

Multiply $\frac{3^{2}}{5^{2}}$ by $1$ .

$r = \sqrt{\frac{3^{2}}{5^{2}} + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.4

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

$r = \sqrt{\frac{9}{5^{2}} + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.5

Raise $5$ to the power of $2$ .

$r = \sqrt{\frac{9}{25} + {(\frac{4}{5})}^{2}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.6

Apply the product rule to $\frac{4}{5}$ .

$r = \sqrt{\frac{9}{25} + \frac{4^{2}}{5^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.7

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

$r = \sqrt{\frac{9}{25} + \frac{16}{5^{2}}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.8

Raise $5$ to the power of $2$ .

$r = \sqrt{\frac{9}{25} + \frac{16}{25}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.9

Combine the numerators over the common denominator.

$r = \sqrt{\frac{9 + 16}{25}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.10

Add $9$ and $16$ .

$r = \sqrt{\frac{25}{25}}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.11

Divide $25$ by $25$ .

$r = \sqrt{1}$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 3.12

Any root of $1$ is $1$ .

$r = 1$

$θ = t a n^{- 1} (\frac{y}{x})$

$r = 1$

$θ = t a n^{- 1} (\frac{y}{x})$

Step 4

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

$r = 1$

$θ = t a n^{- 1} (\frac{\frac{4}{5}}{- \frac{3}{5}})$

Step 5

The inverse tangent of $- \frac{4}{3}$ is $θ = 126.86989764 °$ .

$r = 1$

$θ = 126.86989764 °$

Step 6

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

$(1, 126.86989764 °)$