Trigonometry Examples

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

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

$\frac{3 x}{2} = - \frac{π}{2}$

Step 2

Solve for $x$ .

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Step 2.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = - π$

Step 2.2

Divide each term in $3 x = - π$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x = - π$ by $3$ .

$\frac{3 x}{3} = \frac{- π}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- π}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- π}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

Step 3

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

$\frac{3 x}{2} = \frac{π}{2}$

Step 4

Solve for $x$ .

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Step 4.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = π$

Step 4.2

Divide each term in $3 x = π$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = π$ by $3$ .

$\frac{3 x}{3} = \frac{π}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{π}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Step 5

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

$(- \frac{π}{3}, \frac{π}{3})$

Step 6

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

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Step 6.1

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{3}$

Step 6.3

Combine $π$ and $\frac{2}{3}$ .

$\frac{π \cdot 2}{3}$

Step 6.4

Move $2$ to the left of $π$ .

$\frac{2 π}{3}$

Step 7

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

$x = \frac{π}{3} + \frac{2 π n}{3}$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 9