Trigonometry Examples

$(0, 0)$ , $(\sqrt{3}, 1)$

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.

$m = \frac{1 - (0)}{\sqrt{3} - (0)}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $0$ .

$m = \frac{1 + 0}{\sqrt{3} - (0)}$

Step 4.1.2

Add $1$ and $0$ .

$m = \frac{1}{\sqrt{3} - (0)}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $- 1$ by $0$ .

$m = \frac{1}{\sqrt{3} + 0}$

Step 4.2.2

Add $\sqrt{3}$ and $0$ .

$m = \frac{1}{\sqrt{3}}$

Step 4.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$m = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$m = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.4.6

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

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Step 4.4.6.1

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

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

$m = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$m = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$m = \frac{\sqrt{3}}{3}$

Step 4.4.6.5

Evaluate the exponent.

$m = \frac{\sqrt{3}}{3}$

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 $- \frac{1}{\frac{\sqrt{3}}{3}}$ .

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{\sqrt{3}}))$

Step 6.2

Multiply $\frac{3}{\sqrt{3}}$ by $1$ .

$m_{perpendicular} = - \frac{3}{\sqrt{3}}$

Step 6.3

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m_{perpendicular} = - (\frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m_{perpendicular} = - \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$m_{perpendicular} = - \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$m_{perpendicular} = - \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$m_{perpendicular} = - \frac{3 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$m_{perpendicular} = - \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 6.4.6

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

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Step 6.4.6.1

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

$m_{perpendicular} = - \frac{3 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 6.4.6.2

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

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 6.4.6.3

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

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.4.6.4

Cancel the common factor of $2$ .

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Step 6.4.6.4.1

Cancel the common factor.

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.4.6.4.2

Rewrite the expression.

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3}$

Step 6.4.6.5

Evaluate the exponent.

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3}$

Step 6.5

Cancel the common factor of $3$ .

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Step 6.5.1

Cancel the common factor.

$m_{perpendicular} = - \frac{3 \sqrt{3}}{3}$

Step 6.5.2

Divide $\sqrt{3}$ by $1$ .

$m_{perpendicular} = - \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$m_{perpendicular} = - \sqrt{3}$

Decimal Form:

$m_{perpendicular} = - 1.73205080 \dots$

Step 8