Precalculus Examples

$(4, \frac{7 π}{12})$ , $(2, \frac{π}{12})$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{(2 - 4)}^{2} + {(\frac{π}{12} - \frac{7 π}{12})}^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the expression.

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Step 3.1.1

Subtract $4$ from $2$ .

$\sqrt{{(- 2)}^{2} + {(\frac{π}{12} - \frac{7 π}{12})}^{2}}$

Step 3.1.2

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

$\sqrt{4 + {(\frac{π}{12} - \frac{7 π}{12})}^{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{4 + {(\frac{π - 7 π}{12})}^{2}}$

Step 3.3

Subtract $7 π$ from $π$ .

$\sqrt{4 + {(\frac{- 6 π}{12})}^{2}}$

Step 3.4

Cancel the common factor of $- 6$ and $12$ .

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Step 3.4.1

Factor $6$ out of $- 6 π$ .

$\sqrt{4 + {(\frac{6 (- π)}{12})}^{2}}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $6$ out of $12$ .

$\sqrt{4 + {(\frac{6 (- π)}{6 (2)})}^{2}}$

Step 3.4.2.2

Cancel the common factor.

$\sqrt{4 + {(\frac{6 (- π)}{6 \cdot 2})}^{2}}$

Step 3.4.2.3

Rewrite the expression.

$\sqrt{4 + {(\frac{- π}{2})}^{2}}$

Step 3.5

Move the negative in front of the fraction.

$\sqrt{4 + {(- \frac{π}{2})}^{2}}$

Step 3.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.6.1

Apply the product rule to $- \frac{π}{2}$ .

$\sqrt{4 + {(- 1)}^{2} {(\frac{π}{2})}^{2}}$

Step 3.6.2

Apply the product rule to $\frac{π}{2}$ .

$\sqrt{4 + {(- 1)}^{2} \frac{π^{2}}{2^{2}}}$

Step 3.7

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

$\sqrt{4 + 1 \frac{π^{2}}{2^{2}}}$

Step 3.8

Multiply $\frac{π^{2}}{2^{2}}$ by $1$ .

$\sqrt{4 + \frac{π^{2}}{2^{2}}}$

Step 3.9

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

$\sqrt{4 + \frac{π^{2}}{4}}$

Step 3.10

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{4 \cdot \frac{4}{4} + \frac{π^{2}}{4}}$

Step 3.11

Combine $4$ and $\frac{4}{4}$ .

$\sqrt{\frac{4 \cdot 4}{4} + \frac{π^{2}}{4}}$

Step 3.12

Simplify the expression.

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Step 3.12.1

Combine the numerators over the common denominator.

$\sqrt{\frac{4 \cdot 4 + π^{2}}{4}}$

Step 3.12.2

Multiply $4$ by $4$ .

$\sqrt{\frac{16 + π^{2}}{4}}$

Step 3.13

Rewrite $\sqrt{\frac{16 + π^{2}}{4}}$ as $\frac{\sqrt{16 + π^{2}}}{\sqrt{4}}$ .

$\frac{\sqrt{16 + π^{2}}}{\sqrt{4}}$

Step 3.14

Simplify the denominator.

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Step 3.14.1

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

$\frac{\sqrt{16 + π^{2}}}{\sqrt{2^{2}}}$

Step 3.14.2

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

$\frac{\sqrt{16 + π^{2}}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{16 + π^{2}}}{2}$

Decimal Form:

$2.54310855 \dots$

Step 5