Trigonometry Examples

$f (x) = 1 - | \cos (2 x) |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 1 - | \cos (2 x) |$ is $(\frac{π}{4} + \frac{π n}{2}, 1 - | \cos (\frac{π}{2} + π n) |)$ .

Tap for more steps...

Step 1.1

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

$\cos (2 x) = 0$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$2 x = \arccos (0)$

Step 1.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.1

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

$2 x = \frac{π}{2}$

Step 1.2.3

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

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

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 1.2.3.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.3.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 1.2.3.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 1.2.4

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.

$2 x = 2 π - \frac{π}{2}$

Step 1.2.5

Solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Simplify.

Tap for more steps...

Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.5.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 1.2.5.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 1.2.5.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 1.2.5.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 1.2.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 1.2.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 1.2.5.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.5.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 1.2.5.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 1.2.6

Find the period of $\cos (2 x)$ .

Tap for more steps...

Step 1.2.6.1

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

$\frac{2 π}{| b |}$

Step 1.2.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.6.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.7

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

$x = \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 1.2.8

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 1.3

Replace the variable $x$ with $\frac{π}{4} + \frac{π n}{2}$ in the expression.

$y = 1 - | \cos (2 (\frac{π}{4} + \frac{π n}{2})) |$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

Apply the distributive property.

$y = 1 - | \cos (2 (\frac{π}{4}) + 2 (\frac{π n}{2})) |$

Step 1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

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

Step 1.5

The absolute value vertex is $(\frac{π}{4} + \frac{π n}{2}, 1 - | \cos (\frac{π}{2} + π n) |)$ .

$(\frac{π}{4} + \frac{π n}{2}, 1 - | \cos (\frac{π}{2} + π n) |)$

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 $(\frac{π}{4} + \frac{π n}{2}, 1 - | \cos (\frac{π}{2} + π n) |),$

$\begin{array}{c} x & y \\ \frac{π}{4} + \frac{π n}{2} & 1 - | \cos (\frac{π}{2} + π n) | \end{array}$

Step 4