Trigonometry Examples

$f (x) = 6 | \cot (\frac{π}{12} x) |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 6 | \cot (\frac{π x}{12}) |$ is $(6 + 12 n, 6 | \cot (\frac{π (1 + 2 n)}{2}) |)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\cot (\frac{π x}{12})$ equal to $0$ . In this case, $\cot (\frac{π x}{12}) = 0$ .

$\cot (\frac{π x}{12}) = 0$

Step 1.2

Solve the equation $\cot (\frac{π x}{12}) = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Take the inverse cotangent of both sides of the equation to extract $x$ from inside the cotangent.

$\frac{π x}{12} = arccot (0)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$\frac{π x}{12} = \frac{π}{2}$

Step 1.2.3

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4

Simplify both sides of the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 1.2.4.1.1.1

Cancel the common factor of $12$ .

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Step 1.2.4.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4.1.1.2

Cancel the common factor of $π$ .

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Step 1.2.4.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} \cdot \frac{π}{2}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify $\frac{12}{π} \cdot \frac{π}{2}$ .

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Step 1.2.4.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.1.1

Factor $2$ out of $12$ .

$x = \frac{2 (6)}{π} \cdot \frac{π}{2}$

Step 1.2.4.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 6}{π} \cdot \frac{π}{2}$

Step 1.2.4.2.1.1.3

Rewrite the expression.

$x = \frac{6}{π} π$

Step 1.2.4.2.1.2

Cancel the common factor of $π$ .

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Step 1.2.4.2.1.2.1

Cancel the common factor.

$x = \frac{6}{π} π$

Step 1.2.4.2.1.2.2

Rewrite the expression.

$x = 6$

Step 1.2.5

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$\frac{π x}{12} = π + \frac{π}{2}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2

Simplify both sides of the equation.

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Step 1.2.6.2.1

Simplify the left side.

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Step 1.2.6.2.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 1.2.6.2.1.1.1

Cancel the common factor of $12$ .

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Step 1.2.6.2.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2.1.1.2

Cancel the common factor of $π$ .

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Step 1.2.6.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} (π + \frac{π}{2})$

Step 1.2.6.2.2

Simplify the right side.

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Step 1.2.6.2.2.1

Simplify $\frac{12}{π} (π + \frac{π}{2})$ .

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Step 1.2.6.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{12}{π} (π \cdot \frac{2}{2} + \frac{π}{2})$

Step 1.2.6.2.2.1.2

Simplify terms.

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Step 1.2.6.2.2.1.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{12}{π} (\frac{π \cdot 2}{2} + \frac{π}{2})$

Step 1.2.6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = \frac{12}{π} \cdot \frac{π \cdot 2 + π}{2}$

Step 1.2.6.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 1.2.6.2.2.1.2.3.1

Factor $2$ out of $12$ .

$x = \frac{2 (6)}{π} \cdot \frac{π \cdot 2 + π}{2}$

Step 1.2.6.2.2.1.2.3.2

Cancel the common factor.

$x = \frac{2 \cdot 6}{π} \cdot \frac{π \cdot 2 + π}{2}$

Step 1.2.6.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{6}{π} (π \cdot 2 + π)$

Step 1.2.6.2.2.1.3

Move $2$ to the left of $π$ .

$x = \frac{6}{π} (2 π + π)$

Step 1.2.6.2.2.1.4

Simplify terms.

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Step 1.2.6.2.2.1.4.1

Add $2 π$ and $π$ .

$x = \frac{6}{π} (3 π)$

Step 1.2.6.2.2.1.4.2

Cancel the common factor of $π$ .

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Step 1.2.6.2.2.1.4.2.1

Factor $π$ out of $3 π$ .

$x = \frac{6}{π} (π \cdot 3)$

Step 1.2.6.2.2.1.4.2.2

Cancel the common factor.

$x = \frac{6}{π} (π \cdot 3)$

Step 1.2.6.2.2.1.4.2.3

Rewrite the expression.

$x = 6 \cdot 3$

Step 1.2.6.2.2.1.4.3

Multiply $6$ by $3$ .

$x = 18$

Step 1.2.7

Find the period of $\cot (\frac{π x}{12})$ .

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Step 1.2.7.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 1.2.7.2

Replace $b$ with $\frac{π}{12}$ in the formula for period.

$\frac{π}{| \frac{π}{12} |}$

Step 1.2.7.3

$\frac{π}{12}$ is approximately $0.26179938$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{12}}$

Step 1.2.7.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{12}{π}$

Step 1.2.7.5

Cancel the common factor of $π$ .

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Step 1.2.7.5.1

Cancel the common factor.

$π \frac{12}{π}$

Step 1.2.7.5.2

Rewrite the expression.

$12$

Step 1.2.8

The period of the $\cot (\frac{π x}{12})$ function is $12$ so values will repeat every $12$ radians in both directions.

$x = 6 + 12 n, 18 + 12 n$ , for any integer $n$

Step 1.2.9

Consolidate the answers.

$x = 6 + 12 n$ , for any integer $n$

Step 1.3

Replace the variable $x$ with $6 + 12 n$ in the expression.

$y = 6 | \cot (\frac{π (6 + 12 n)}{12}) |$

Step 1.4

Cancel the common factor of $6 + 12 n$ and $12$ .

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Step 1.4.1

Factor $6$ out of $π (6 + 12 n)$ .

$y = 6 | \cot (\frac{6 (π (1 + 2 n))}{12}) |$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $6$ out of $12$ .

$y = 6 | \cot (\frac{6 (π (1 + 2 n))}{6 (2)}) |$

Step 1.4.2.2

Cancel the common factor.

$y = 6 | \cot (\frac{6 (π (1 + 2 n))}{6 \cdot 2}) |$

Step 1.4.2.3

Rewrite the expression.

$y = 6 | \cot (\frac{π (1 + 2 n)}{2}) |$

Step 1.5

The absolute value vertex is $(6 + 12 n, 6 | \cot (\frac{π (1 + 2 n)}{2}) |)$ .

$(6 + 12 n, 6 | \cot (\frac{π (1 + 2 n)}{2}) |)$

Step 2

Find the domain for $f (x) = 6 | \cot (\frac{π x}{12}) |$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

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Step 2.1

Set the argument in $\cot (\frac{π x}{12})$ equal to $π n$ to find where the expression is undefined.

$\frac{π x}{12} = π n$ , for any integer $n$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (π n)$

Step 2.2.2

Simplify both sides of the equation.

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Step 2.2.2.1

Simplify the left side.

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Step 2.2.2.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 2.2.2.1.1.1

Cancel the common factor of $12$ .

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Step 2.2.2.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (π n)$

Step 2.2.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} (π n)$

Step 2.2.2.1.1.2

Cancel the common factor of $π$ .

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Step 2.2.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} (π n)$

Step 2.2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} (π n)$

Step 2.2.2.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} (π n)$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

Cancel the common factor of $π$ .

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Step 2.2.2.2.1.1

Factor $π$ out of $π n$ .

$x = \frac{12}{π} (π (n))$

Step 2.2.2.2.1.2

Cancel the common factor.

$x = \frac{12}{π} (π n)$

Step 2.2.2.2.1.3

Rewrite the expression.

$x = 12 n$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq 12 n}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq 12 n}$ , for any integer $n$

Step 3

The absolute value can be graphed using the points around the vertex $(6 + 12 n, 6 | \cot (\frac{π (1 + 2 n)}{2}) |),$

$\begin{array}{c} x & y \\ 6 + 12 n & 6 | \cot (\frac{π (1 + 2 n)}{2}) | \end{array}$

Step 4