Trigonometry Examples

$(\frac{3}{10}, - 1)$ , $(- \frac{1}{5}, - \frac{1}{5})$

Step 1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- \frac{1}{5} - (- 1)}{- \frac{1}{5} - (\frac{3}{10})}$

Step 4

Simplify.

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Step 4.1

Multiply the numerator and denominator of the fraction by $10$ .

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Step 4.1.1

Multiply $\frac{- \frac{1}{5} - (- 1)}{- \frac{1}{5} - \frac{3}{10}}$ by $\frac{10}{10}$ .

$m = \frac{10}{10} \cdot \frac{- \frac{1}{5} - (- 1)}{- \frac{1}{5} - \frac{3}{10}}$

Step 4.1.2

Combine.

$m = \frac{10 (- \frac{1}{5} - (- 1))}{10 (- \frac{1}{5} - \frac{3}{10})}$

Step 4.2

Apply the distributive property.

$m = \frac{10 (- \frac{1}{5}) + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3

Simplify by cancelling.

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Step 4.3.1

Cancel the common factor of $5$ .

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Step 4.3.1.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$m = \frac{10 (\frac{- 1}{5}) + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.1.2

Factor $5$ out of $10$ .

$m = \frac{5 (2) (\frac{- 1}{5}) + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.1.3

Cancel the common factor.

$m = \frac{5 \cdot (2 (\frac{- 1}{5})) + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.1.4

Rewrite the expression.

$m = \frac{2 \cdot - 1 + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.2

Multiply $2$ by $- 1$ .

$m = \frac{- 2 + 10 (- (- 1))}{10 (- \frac{1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.3

Cancel the common factor of $5$ .

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Step 4.3.3.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$m = \frac{- 2 + 10 (- (- 1))}{10 (\frac{- 1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.3.2

Factor $5$ out of $10$ .

$m = \frac{- 2 + 10 (- (- 1))}{5 (2) (\frac{- 1}{5}) + 10 (- \frac{3}{10})}$

Step 4.3.3.3

Cancel the common factor.

$m = \frac{- 2 + 10 (- (- 1))}{5 \cdot (2 (\frac{- 1}{5})) + 10 (- \frac{3}{10})}$

Step 4.3.3.4

Rewrite the expression.

$m = \frac{- 2 + 10 (- (- 1))}{2 \cdot - 1 + 10 (- \frac{3}{10})}$

Step 4.3.4

Multiply $2$ by $- 1$ .

$m = \frac{- 2 + 10 (- (- 1))}{- 2 + 10 (- \frac{3}{10})}$

Step 4.3.5

Cancel the common factor of $10$ .

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Step 4.3.5.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$m = \frac{- 2 + 10 (- (- 1))}{- 2 + 10 (\frac{- 3}{10})}$

Step 4.3.5.2

Cancel the common factor.

$m = \frac{- 2 + 10 (- (- 1))}{- 2 + 10 (\frac{- 3}{10})}$

Step 4.3.5.3

Rewrite the expression.

$m = \frac{- 2 + 10 (- (- 1))}{- 2 - 3}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Multiply $10 (- (- 1))$ .

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Step 4.4.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 2 + 10 \cdot 1}{- 2 - 3}$

Step 4.4.1.2

Multiply $10$ by $1$ .

$m = \frac{- 2 + 10}{- 2 - 3}$

Step 4.4.2

Add $- 2$ and $10$ .

$m = \frac{8}{- 2 - 3}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Subtract $3$ from $- 2$ .

$m = \frac{8}{- 5}$

Step 4.5.2

Move the negative in front of the fraction.

$m = - \frac{8}{5}$

Step 5

The slope of a perpendicular line is the negative reciprocal of the slope of the line that passes through the two given points.

$m_{perpendicular} = - \frac{1}{m}$

Step 6

Simplify $- \frac{1}{- \frac{8}{5}}$ .

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Step 6.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{8}{5}}$

Step 6.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{8}{5}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{8})$

Step 6.3

Multiply $\frac{5}{8}$ by $1$ .

$m_{perpendicular} = \frac{5}{8}$

Step 6.4

Multiply $- - \frac{5}{8}$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{8})$

Step 6.4.2

Multiply $\frac{5}{8}$ by $1$ .

$m_{perpendicular} = \frac{5}{8}$

Step 7