Trigonometry Examples

$y = 3 \tan (\frac{x}{4})$

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 $y = 3 \tan (\frac{x}{4})$ . 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 $y = 3 \tan (\frac{x}{4})$ .

$\frac{x}{4} = - \frac{π}{2}$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 (- \frac{π}{2})$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 (- \frac{π}{2})$

Step 2.2.1.1.2

Rewrite the expression.

$x = 4 (- \frac{π}{2})$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $4 (- \frac{π}{2})$ .

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Step 2.2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = 4 \frac{- π}{2}$

Step 2.2.2.1.1.2

Factor $2$ out of $4$ .

$x = 2 (2) \frac{- π}{2}$

Step 2.2.2.1.1.3

Cancel the common factor.

$x = 2 \cdot 2 \frac{- π}{2}$

Step 2.2.2.1.1.4

Rewrite the expression.

$x = 2 (- π)$

Step 2.2.2.1.2

Multiply $- 1$ by $2$ .

$x = - 2 π$

Step 3

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

$\frac{x}{4} = \frac{π}{2}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 \frac{π}{2}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 \frac{π}{2}$

Step 4.2.1.1.2

Rewrite the expression.

$x = 4 \frac{π}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{π}{2}$

Step 4.2.2.1.2

Cancel the common factor.

$x = 2 \cdot 2 \frac{π}{2}$

Step 4.2.2.1.3

Rewrite the expression.

$x = 2 π$

Step 5

The basic period for $y = 3 \tan (\frac{x}{4})$ will occur at $(- 2 π, 2 π)$ , where $- 2 π$ and $2 π$ are vertical asymptotes.

$(- 2 π, 2 π)$

Step 6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 6.1

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{4}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 4$

Step 6.3

Move $4$ to the left of $π$ .

$4 π$

Step 7

The vertical asymptotes for $y = 3 \tan (\frac{x}{4})$ occur at $- 2 π$ , $2 π$ , and every $4 π n$ , where $n$ is an integer.

$x = 2 π + 4 π n$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π + 4 π n$ where $n$ is an integer

Step 9