Trigonometry Examples

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

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

$\frac{x}{3} = - \frac{π}{2}$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (- \frac{π}{2})$

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 $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (- \frac{π}{2})$

Step 2.2.1.1.2

Rewrite the expression.

$x = 3 (- \frac{π}{2})$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $3 (- \frac{π}{2})$ .

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Step 2.2.2.1.1

Multiply $3 (- \frac{π}{2})$ .

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Step 2.2.2.1.1.1

Multiply $- 1$ by $3$ .

$x = - 3 \frac{π}{2}$

Step 2.2.2.1.1.2

Combine $- 3$ and $\frac{π}{2}$ .

$x = \frac{- 3 π}{2}$

Step 2.2.2.1.2

Move the negative in front of the fraction.

$x = - \frac{3 π}{2}$

Step 3

Set the inside of the tangent function $\frac{x}{3}$ equal to $\frac{π}{2}$ .

$\frac{x}{3} = \frac{π}{2}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \frac{π}{2}$

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 $3$ .

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Step 4.2.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \frac{π}{2}$

Step 4.2.1.1.2

Rewrite the expression.

$x = 3 \frac{π}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Combine $3$ and $\frac{π}{2}$ .

$x = \frac{3 π}{2}$

Step 5

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

$(- \frac{3 π}{2}, \frac{3 π}{2})$

Step 6

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

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Step 6.1

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Step 6.3

Move $3$ to the left of $π$ .

$3 π$

Step 7

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

$x = \frac{3 π}{2} + 3 π n$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{3 π}{2} + 3 π n$ where $n$ is an integer

Step 9