Trigonometry Examples

$y = 2 \tan (\frac{π}{6} x)$

Step 1

Combine $\frac{π}{6}$ and $x$ .

$y = 2 \tan (\frac{π x}{6})$

Step 2

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 $y = 2 \tan (\frac{π x}{6})$ . 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 $y = 2 \tan (\frac{π x}{6})$ .

$\frac{π x}{6} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (- \frac{π}{2})$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 3.2.1.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (- \frac{π}{2})$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} (- \frac{π}{2})$

Step 3.2.1.1.2

Cancel the common factor of $π$ .

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Step 3.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} (- \frac{π}{2})$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} (- \frac{π}{2})$

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} (- \frac{π}{2})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{6}{π} (- \frac{π}{2})$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

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

$x = \frac{6}{π} \cdot \frac{- π}{2}$

Step 3.2.2.1.1.2

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{π} \cdot \frac{- π}{2}$

Step 3.2.2.1.1.3

Cancel the common factor.

$x = \frac{2 \cdot 3}{π} \cdot \frac{- π}{2}$

Step 3.2.2.1.1.4

Rewrite the expression.

$x = \frac{3}{π} (- π)$

Step 3.2.2.1.2

Cancel the common factor of $π$ .

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Step 3.2.2.1.2.1

Factor $π$ out of $- π$ .

$x = \frac{3}{π} (π \cdot - 1)$

Step 3.2.2.1.2.2

Cancel the common factor.

$x = \frac{3}{π} (π \cdot - 1)$

Step 3.2.2.1.2.3

Rewrite the expression.

$x = 3 \cdot - 1$

Step 3.2.2.1.3

Multiply $3$ by $- 1$ .

$x = - 3$

Step 4

Set the inside of the tangent function $\frac{π x}{6}$ equal to $\frac{π}{2}$ .

$\frac{π x}{6} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 5.2.1.1.1

Cancel the common factor of $6$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2.1.1.2

Cancel the common factor of $π$ .

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Step 5.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} \cdot \frac{π}{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{6}{π} \cdot \frac{π}{2}$ .

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Step 5.2.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{π} \cdot \frac{π}{2}$

Step 5.2.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 3}{π} \cdot \frac{π}{2}$

Step 5.2.2.1.1.3

Rewrite the expression.

$x = \frac{3}{π} π$

Step 5.2.2.1.2

Cancel the common factor of $π$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

$x = \frac{3}{π} π$

Step 5.2.2.1.2.2

Rewrite the expression.

$x = 3$

Step 6

The basic period for $y = 2 \tan (\frac{π x}{6})$ will occur at $(- 3, 3)$ , where $- 3$ and $3$ are vertical asymptotes.

$(- 3, 3)$

Step 7

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

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Step 7.1

$\frac{π}{6}$ is approximately $0.52359877$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{6}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{6}{π}$

Step 7.3

Cancel the common factor of $π$ .

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Step 7.3.1

Cancel the common factor.

$π \frac{6}{π}$

Step 7.3.2

Rewrite the expression.

$6$

Step 8

The vertical asymptotes for $y = 2 \tan (\frac{π x}{6})$ occur at $- 3$ , $3$ , and every $6 n$ , where $n$ is an integer.

$x = 3 + 6 n$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 3 + 6 n$ where $n$ is an integer

Step 10