Trigonometry Examples

$f (x) = 4 | \sin (x) |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 4 | \sin (x) |$ is $(π n, 4 | \sin (π n) |)$ .

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Step 1.1

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

$\sin (x) = 0$

Step 1.2

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

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Step 1.2.1

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

$x = \arcsin (0)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 1.2.3

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

$x = π - 0$

Step 1.2.4

Subtract $0$ from $π$ .

$x = π$

Step 1.2.5

Find the period of $\sin (x)$ .

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Step 1.2.5.1

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

$\frac{2 π}{| b |}$

Step 1.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.6

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

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

Step 1.2.7

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 1.3

Replace the variable $x$ with $π n$ in the expression.

$y = 4 | \sin (π n) |$

Step 1.4

The absolute value vertex is $(π n, 4 | \sin (π n) |)$ .

$(π n, 4 | \sin (π n) |)$

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 $(π n, 4 | \sin (π n) |),$

$\begin{array}{c} x & y \\ π n & 4 | \sin (π n) | \end{array}$

Step 4