Trigonometry Examples

$f (x) = | 2 \cos (\frac{π x}{2}) |$

Step 1

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

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\cos (\frac{π x}{2})$ equal to $0$ . In this case, $\cos (\frac{π x}{2}) = 0$ .

$\cos (\frac{π x}{2}) = 0$

Step 1.2

Solve the equation $\cos (\frac{π x}{2}) = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

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

$\frac{π x}{2} = \arccos (0)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

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

$\frac{π x}{2} = \frac{π}{2}$

Step 1.2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$π x = π$

Step 1.2.4

Divide each term in $π x = π$ by $π$ and simplify.

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Step 1.2.4.1

Divide each term in $π x = π$ by $π$ .

$\frac{π x}{π} = \frac{π}{π}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $π$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{π}$

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{π}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Cancel the common factor of $π$ .

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Step 1.2.4.3.1.1

Cancel the common factor.

$x = \frac{π}{π}$

Step 1.2.4.3.1.2

Rewrite the expression.

$x = 1$

Step 1.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$\frac{π x}{2} = 2 π - \frac{π}{2}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Multiply both sides of the equation by $\frac{2}{π}$ .

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

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 $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 1.2.6.2.1.1.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1.1.1

Cancel the common factor.

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

Step 1.2.6.2.1.1.1.2

Rewrite the expression.

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

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 $π x$ .

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

Step 1.2.6.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.6.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} (2 π - \frac{π}{2})$

Step 1.2.6.2.2

Simplify the right side.

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Step 1.2.6.2.2.1

Simplify $\frac{2}{π} (2 π - \frac{π}{2})$ .

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Step 1.2.6.2.2.1.1

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

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

Step 1.2.6.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.6.2.2.1.3

Combine the numerators over the common denominator.

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

Step 1.2.6.2.2.1.4

Cancel the common factor of $2$ .

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Step 1.2.6.2.2.1.4.1

Cancel the common factor.

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

Step 1.2.6.2.2.1.4.2

Rewrite the expression.

$x = \frac{1}{π} (2 π \cdot 2 - π)$

Step 1.2.6.2.2.1.5

Multiply $2$ by $2$ .

$x = \frac{1}{π} (4 π - π)$

Step 1.2.6.2.2.1.6

Subtract $π$ from $4 π$ .

$x = \frac{1}{π} (3 π)$

Step 1.2.6.2.2.1.7

Cancel the common factor of $π$ .

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Step 1.2.6.2.2.1.7.1

Factor $π$ out of $3 π$ .

$x = \frac{1}{π} (π \cdot 3)$

Step 1.2.6.2.2.1.7.2

Cancel the common factor.

$x = \frac{1}{π} (π \cdot 3)$

Step 1.2.6.2.2.1.7.3

Rewrite the expression.

$x = 3$

Step 1.2.7

Find the period of $\cos (\frac{π x}{2})$ .

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Step 1.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.7.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 1.2.7.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 1.2.7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 1.2.7.5

Cancel the common factor of $π$ .

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Step 1.2.7.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 1.2.7.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 1.2.7.5.3

Rewrite the expression.

$2 \cdot 2$

Step 1.2.7.6

Multiply $2$ by $2$ .

$4$

Step 1.2.8

The period of the $\cos (\frac{π x}{2})$ function is $4$ so values will repeat every $4$ radians in both directions.

$x = 1 + 4 n, 3 + 4 n$ , for any integer $n$

Step 1.2.9

Consolidate the answers.

$x = 1 + 2 n$ , for any integer $n$

Step 1.3

Replace the variable $x$ with $1 + 2 n$ in the expression.

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

Step 1.4

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

$(1 + 2 n, 2 | \cos (\frac{π (1 + 2 n)}{2}) |)$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

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

Step 4