Trigonometry Examples

$(\frac{1}{4}, 3)$ , $(3, 3)$

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.

$m = \frac{3 - (3)}{3 - (\frac{1}{4})}$

Step 4

Simplify.

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Step 4.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 4.1.1

Multiply $\frac{3 - (3)}{3 - \frac{1}{4}}$ by $\frac{4}{4}$ .

$m = \frac{4}{4} \cdot \frac{3 - (3)}{3 - \frac{1}{4}}$

Step 4.1.2

Combine.

$m = \frac{4 (3 - (3))}{4 (3 - \frac{1}{4})}$

Step 4.2

Apply the distributive property.

$m = \frac{4 \cdot 3 + 4 (- (3))}{4 \cdot 3 + 4 (- \frac{1}{4})}$

Step 4.3

Cancel the common factor of $4$ .

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Step 4.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$m = \frac{4 \cdot 3 + 4 (- (3))}{4 \cdot 3 + 4 (\frac{- 1}{4})}$

Step 4.3.2

Cancel the common factor.

$m = \frac{4 \cdot 3 + 4 (- (3))}{4 \cdot 3 + 4 (\frac{- 1}{4})}$

Step 4.3.3

Rewrite the expression.

$m = \frac{4 \cdot 3 + 4 (- (3))}{4 \cdot 3 - 1}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Multiply $4$ by $3$ .

$m = \frac{12 + 4 (- (3))}{4 \cdot 3 - 1}$

Step 4.4.2

Multiply $4 (- (3))$ .

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Step 4.4.2.1

Multiply $- 1$ by $3$ .

$m = \frac{12 + 4 \cdot - 3}{4 \cdot 3 - 1}$

Step 4.4.2.2

Multiply $4$ by $- 3$ .

$m = \frac{12 - 12}{4 \cdot 3 - 1}$

Step 4.4.3

Subtract $12$ from $12$ .

$m = \frac{0}{4 \cdot 3 - 1}$

Step 4.5

Simplify the denominator.

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Step 4.5.1

Multiply $4$ by $3$ .

$m = \frac{0}{12 - 1}$

Step 4.5.2

Subtract $1$ from $12$ .

$m = \frac{0}{11}$

Step 4.6

Divide $0$ by $11$ .

$m = 0$

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

The slope of the perpendicular line is $- \frac{1}{0}$ .

$m_{perpendicular} = - \frac{1}{0}$

Step 7

The slope of a line perpendicular to a horizontal line is undefined.

Undefined Slope

Step 8