Trigonometry Examples

$y = \tan (\frac{3 π}{4} θ)$

Step 1

Combine $\frac{3 π}{4}$ and $x$ .

$y = \tan (\frac{3 π x}{4})$

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

$\frac{3 π x}{4} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{4}{3 π}$ .

$\frac{4}{3 π} \cdot \frac{3 π x}{4} = \frac{4}{3 π} (- \frac{π}{2})$

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 $\frac{4}{3 π} \cdot \frac{3 π x}{4}$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{4}{3 π} \cdot \frac{3 π x}{4} = \frac{4}{3 π} (- \frac{π}{2})$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3 π} (3 π x) = \frac{4}{3 π} (- \frac{π}{2})$

Step 3.2.1.1.2

Cancel the common factor of $3 π$ .

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Step 3.2.1.1.2.1

Factor $3 π$ out of $3 π x$ .

$\frac{1}{3 π} (3 π (x)) = \frac{4}{3 π} (- \frac{π}{2})$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3 π} (3 π x) = \frac{4}{3 π} (- \frac{π}{2})$

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3 π} (- \frac{π}{2})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{4}{3 π} (- \frac{π}{2})$ .

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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.

$x = \frac{4}{3 π} \cdot \frac{- π}{2}$

Step 3.2.2.1.1.2

Factor $2$ out of $4$ .

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

Step 3.2.2.1.1.3

Cancel the common factor.

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

Step 3.2.2.1.1.4

Rewrite the expression.

$x = \frac{2}{3 π} (- π)$

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 $3 π$ .

$x = \frac{2}{π \cdot 3} (- π)$

Step 3.2.2.1.2.2

Factor $π$ out of $- π$ .

$x = \frac{2}{π \cdot 3} (π \cdot - 1)$

Step 3.2.2.1.2.3

Cancel the common factor.

$x = \frac{2}{π \cdot 3} (π \cdot - 1)$

Step 3.2.2.1.2.4

Rewrite the expression.

$x = \frac{2}{3} \cdot - 1$

Step 3.2.2.1.3

Combine $\frac{2}{3}$ and $- 1$ .

$x = \frac{2 \cdot - 1}{3}$

Step 3.2.2.1.4

Simplify the expression.

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Step 3.2.2.1.4.1

Multiply $2$ by $- 1$ .

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

Step 3.2.2.1.4.2

Move the negative in front of the fraction.

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

Step 4

Set the inside of the tangent function $\frac{3 π x}{4}$ equal to $\frac{π}{2}$ .

$\frac{3 π x}{4} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $\frac{4}{3 π}$ .

$\frac{4}{3 π} \cdot \frac{3 π x}{4} = \frac{4}{3 π} \cdot \frac{π}{2}$

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 $\frac{4}{3 π} \cdot \frac{3 π x}{4}$ .

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Step 5.2.1.1.1

Cancel the common factor of $4$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{4}{3 π} \cdot \frac{3 π x}{4} = \frac{4}{3 π} \cdot \frac{π}{2}$

Step 5.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3 π} (3 π x) = \frac{4}{3 π} \cdot \frac{π}{2}$

Step 5.2.1.1.2

Cancel the common factor of $3 π$ .

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Step 5.2.1.1.2.1

Factor $3 π$ out of $3 π x$ .

$\frac{1}{3 π} (3 π (x)) = \frac{4}{3 π} \cdot \frac{π}{2}$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3 π} (3 π x) = \frac{4}{3 π} \cdot \frac{π}{2}$

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3 π} \cdot \frac{π}{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{4}{3 π} \cdot \frac{π}{2}$ .

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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 $4$ .

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

Step 5.2.2.1.1.2

Cancel the common factor.

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

Step 5.2.2.1.1.3

Rewrite the expression.

$x = \frac{2}{3 π} π$

Step 5.2.2.1.2

Cancel the common factor of $π$ .

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Step 5.2.2.1.2.1

Factor $π$ out of $3 π$ .

$x = \frac{2}{π \cdot 3} π$

Step 5.2.2.1.2.2

Cancel the common factor.

$x = \frac{2}{π \cdot 3} π$

Step 5.2.2.1.2.3

Rewrite the expression.

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

Step 6

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

$(- \frac{2}{3}, \frac{2}{3})$

Step 7

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

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Step 7.1

$\frac{3 π}{4}$ is approximately $2.35619449$ which is positive so remove the absolute value

$\frac{π}{\frac{3 π}{4}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{3 π}$

Step 7.3

Cancel the common factor of $π$ .

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Step 7.3.1

Factor $π$ out of $3 π$ .

$π \frac{4}{π \cdot 3}$

Step 7.3.2

Cancel the common factor.

$π \frac{4}{π \cdot 3}$

Step 7.3.3

Rewrite the expression.

$\frac{4}{3}$

Step 8

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

$x = \frac{2}{3} + \frac{4 n}{3}$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{2}{3} + \frac{4 n}{3}$ where $n$ is an integer

Step 10