Trigonometry Examples

$y = \tan (\frac{2}{3} θ)$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$y = \tan (\frac{2 x}{3})$

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 $y = \tan (\frac{2 x}{3})$ . 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 $y = \tan (\frac{2 x}{3})$ .

$\frac{2 x}{3} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

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 $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

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

Step 3.2.1.1.1.2

Rewrite the expression.

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 3.2.1.1.2.2

Cancel the common factor.

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

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} (- \frac{π}{2})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Multiply $\frac{3}{2}$ by $\frac{π}{2}$ .

$x = - \frac{3 π}{2 \cdot 2}$

Step 3.2.2.1.2

Multiply $2$ by $2$ .

$x = - \frac{3 π}{4}$

Step 4

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

$\frac{2 x}{3} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

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 $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 5.2.1.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.1.2

Rewrite the expression.

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 5.2.1.1.2.2

Cancel the common factor.

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

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} \cdot \frac{π}{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $\frac{3}{2} \cdot \frac{π}{2}$ .

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Step 5.2.2.1.1

Multiply $\frac{3}{2}$ by $\frac{π}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 5.2.2.1.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 6

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

$(- \frac{3 π}{4}, \frac{3 π}{4})$

Step 7

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

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Step 7.1

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{π}{\frac{2}{3}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{3}{2}$

Step 7.3

Combine $π$ and $\frac{3}{2}$ .

$\frac{π \cdot 3}{2}$

Step 7.4

Move $3$ to the left of $π$ .

$\frac{3 π}{2}$

Step 8

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

$x = \frac{3 π}{4} + \frac{3 π n}{2}$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 10