Trigonometry Examples

$f (x) = \tan (3 x - \frac{π}{2})$

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

$3 x - \frac{π}{2} = - \frac{π}{2}$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1.1

Add $\frac{π}{2}$ to both sides of the equation.

$3 x = - \frac{π}{2} + \frac{π}{2}$

Step 2.1.2

Combine the numerators over the common denominator.

$3 x = \frac{- π + π}{2}$

Step 2.1.3

Add $- π$ and $π$ .

$3 x = \frac{0}{2}$

Step 2.1.4

Divide $0$ by $2$ .

$3 x = 0$

Step 2.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

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.

$\frac{3 x}{3} = \frac{0}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 3

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

$3 x - \frac{π}{2} = \frac{π}{2}$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1.1

Add $\frac{π}{2}$ to both sides of the equation.

$3 x = \frac{π}{2} + \frac{π}{2}$

Step 4.1.2

Combine the numerators over the common denominator.

$3 x = \frac{π + π}{2}$

Step 4.1.3

Add $π$ and $π$ .

$3 x = \frac{2 π}{2}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$3 x = \frac{2 π}{2}$

Step 4.1.4.2

Divide $π$ by $1$ .

$3 x = π$

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 $y = \tan (3 x - \frac{π}{2})$ will occur at $(0, \frac{π}{3})$ , where $0$ and $\frac{π}{3}$ are vertical asymptotes.

$(0, \frac{π}{3})$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 7

The vertical asymptotes for $y = \tan (3 x - \frac{π}{2})$ occur at $0$ , $\frac{π}{3}$ , and every $x = 0 + \frac{π n}{3}$ , where $n$ is an integer.

$x = 0 + \frac{π n}{3}$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 0 + \frac{π n}{3}$ where $n$ is an integer

Step 9