Trigonometry Examples

$y = \frac{| x |}{x} \cdot \tan (x)$

Step 1

Find the domain for $y = \frac{| x | \tan (x)}{x}$ 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 1.1

Set the denominator in $\frac{| x | \tan (x)}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.2

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 1.3

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

Set-Builder Notation:

${x | x \neq 0, x \neq \frac{π}{2} + π n}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq 0, x \neq \frac{π}{2} + π n}$ , for any integer $n$

Step 2

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

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Step 2.1

Substitute the $x$ value $- 2$ into $f (x) = \frac{| x | \tan (x)}{x}$ . In this case, the point is $(- 2, 0.03492076)$ .

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Step 2.1.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{| - 2 | \tan (- 2)}{- 2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

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

$f (- 2) = \frac{2 \tan (- 2)}{- 2}$

Step 2.1.2.1.2

Evaluate $\tan (- 2)$ .

$f (- 2) = \frac{2 \cdot - 0.03492076}{- 2}$

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Multiply $2$ by $- 0.03492076$ .

$f (- 2) = \frac{- 0.06984153}{- 2}$

Step 2.1.2.2.2

Divide $- 0.06984153$ by $- 2$ .

$f (- 2) = 0.03492076$

Step 2.1.2.3

The final answer is $0.03492076$ .

$y = 0.03492076$

Step 2.2

Substitute the $x$ value $- 1$ into $f (x) = \frac{| x | \tan (x)}{x}$ . In this case, the point is $(- 1, 0.01745506)$ .

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Step 2.2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{| - 1 | \tan (- 1)}{- 1}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Move the negative one from the denominator of $\frac{| - 1 | \tan (- 1)}{- 1}$ .

$f (- 1) = - 1 \cdot (| - 1 | \tan (- 1))$

Step 2.2.2.2

Rewrite $- 1 \cdot (| - 1 | \tan (- 1))$ as $- (| - 1 | \tan (- 1))$ .

$f (- 1) = - (| - 1 | \tan (- 1))$

Step 2.2.2.3

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

$f (- 1) = - (1 \tan (- 1))$

Step 2.2.2.4

Multiply $\tan (- 1)$ by $1$ .

$f (- 1) = - \tan (- 1)$

Step 2.2.2.5

Evaluate $\tan (- 1)$ .

$f (- 1) = 0.01745506$

Step 2.2.2.6

The final answer is $0.01745506$ .

$y = 0.01745506$

Step 2.3

Substitute the $x$ value $2$ into $f (x) = \frac{| x | \tan (x)}{x}$ . In this case, the point is $(2, 0.03492076)$ .

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Step 2.3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{| 2 | \tan (2)}{2}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify the numerator.

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Step 2.3.2.1.1

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

$f (2) = \frac{2 \tan (2)}{2}$

Step 2.3.2.1.2

Evaluate $\tan (2)$ .

$f (2) = \frac{2 \cdot 0.03492076}{2}$

Step 2.3.2.2

Simplify the expression.

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Step 2.3.2.2.1

Multiply $2$ by $0.03492076$ .

$f (2) = \frac{0.06984153}{2}$

Step 2.3.2.2.2

Divide $0.06984153$ by $2$ .

$f (2) = 0.03492076$

Step 2.3.2.3

The final answer is $0.03492076$ .

$y = 0.03492076$

Step 2.4

The absolute value can be graphed using the points around the vertex $(- 2, 0.03), (- 1, 0.02), (1, 0.02), (2, 0.03)$

$\begin{array}{c} x & y \\ - 2 & 0.03 \\ - 1 & 0.02 \\ 1 & 0.02 \\ 2 & 0.03 \end{array}$

Step 3