Trigonometry Examples

$f (x) = \tan (4 x)$

Step 1

For any $y = \tan (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = \tan (x)$ , $(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for $y = \tan (4 x)$ . Set the inside of the tangent function, $b x + c$ , for $y = a \tan (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = \tan (4 x)$ .

$4 x = - \frac{π}{2}$

Step 2

Divide each term in $4 x = - \frac{π}{2}$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x = - \frac{π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{- \frac{π}{2}}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{π}{2}}{4}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{4}$

Step 2.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{4}$ .

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Step 2.3.2.1

Multiply $\frac{1}{4}$ by $\frac{π}{2}$ .

$x = - \frac{π}{4 \cdot 2}$

Step 2.3.2.2

Multiply $4$ by $2$ .

$x = - \frac{π}{8}$

Step 3

Set the inside of the tangent function $4 x$ equal to $\frac{π}{2}$ .

$4 x = \frac{π}{2}$

Step 4

Divide each term in $4 x = \frac{π}{2}$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 x = \frac{π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{4}$

Step 4.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

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Step 4.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Step 4.3.2.2

Multiply $2$ by $4$ .

$x = \frac{π}{8}$

Step 5

The basic period for $y = \tan (4 x)$ will occur at $(- \frac{π}{8}, \frac{π}{8})$ , where $- \frac{π}{8}$ and $\frac{π}{8}$ are vertical asymptotes.

$(- \frac{π}{8}, \frac{π}{8})$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{π}{4}$

Step 7

The vertical asymptotes for $y = \tan (4 x)$ occur at $- \frac{π}{8}$ , $\frac{π}{8}$ , and every $\frac{π n}{4}$ , where $n$ is an integer.

$x = \frac{π}{8} + \frac{π n}{4}$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{8} + \frac{π n}{4}$ where $n$ is an integer

Step 9