Trigonometry Examples

y = \tan (3 x)

$y = \tan (3 x)$

Step 1

Find the asymptotes.

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Step 1.1

For any

y = \tan (x)

$y = \tan (x)$ , vertical asymptotes occur at

x = \frac{π}{2} + n π

$x = \frac{π}{2} + n π$ , where

n

$n$ is an integer. Use the basic period for

y = \tan (x)

$y = \tan (x)$ ,

(- \frac{π}{2}, \frac{π}{2})

$(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for

y = \tan (3 x)

$y = \tan (3 x)$ . Set the inside of the tangent function,

b x + c

$b x + c$ , for

y = a \tan (b x + c) + d

$y = a \tan (b x + c) + d$ equal to

- \frac{π}{2}

$- \frac{π}{2}$ to find where the vertical asymptote occurs for

y = \tan (3 x)

$y = \tan (3 x)$ .

3 x = - \frac{π}{2}

$3 x = - \frac{π}{2}$

Step 1.2

Divide each term in

3 x = - \frac{π}{2}

$3 x = - \frac{π}{2}$ by

3

$3$ and simplify.

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Step 1.2.1

Divide each term in

3 x = - \frac{π}{2}

$3 x = - \frac{π}{2}$ by

3

$3$ .

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

3

$3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

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

Step 1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- \frac{π}{2}}{3}

$x = \frac{- \frac{π}{2}}{3}$

x = \frac{- \frac{π}{2}}{3}

$x = \frac{- \frac{π}{2}}{3}$

x = \frac{- \frac{π}{2}}{3}

$x = \frac{- \frac{π}{2}}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Multiply the numerator by the reciprocal of the denominator.

x = - \frac{π}{2} \cdot \frac{1}{3}

$x = - \frac{π}{2} \cdot \frac{1}{3}$

Step 1.2.3.2

Multiply

- \frac{π}{2} \cdot \frac{1}{3}

$- \frac{π}{2} \cdot \frac{1}{3}$ .

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Step 1.2.3.2.1

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

\frac{π}{2}

$\frac{π}{2}$ .

x = - \frac{π}{3 \cdot 2}

$x = - \frac{π}{3 \cdot 2}$

Step 1.2.3.2.2

Multiply

3

$3$ by

2

$2$ .

x = - \frac{π}{6}

$x = - \frac{π}{6}$

x = - \frac{π}{6}

$x = - \frac{π}{6}$

x = - \frac{π}{6}

$x = - \frac{π}{6}$

x = - \frac{π}{6}

$x = - \frac{π}{6}$

Step 1.3

Set the inside of the tangent function

3 x

$3 x$ equal to

\frac{π}{2}

$\frac{π}{2}$ .

3 x = \frac{π}{2}

$3 x = \frac{π}{2}$

Step 1.4

Divide each term in

3 x = \frac{π}{2}

$3 x = \frac{π}{2}$ by

3

$3$ and simplify.

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Step 1.4.1

Divide each term in

3 x = \frac{π}{2}

$3 x = \frac{π}{2}$ by

3

$3$ .

\frac{3 x}{3} = \frac{\frac{π}{2}}{3}

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

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of

3

$3$ .

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Step 1.4.2.1.1

Cancel the common factor.

\frac{3 x}{3} = \frac{\frac{π}{2}}{3}

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

Step 1.4.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{π}{2}}{3}

$x = \frac{\frac{π}{2}}{3}$

x = \frac{\frac{π}{2}}{3}

$x = \frac{\frac{π}{2}}{3}$

x = \frac{\frac{π}{2}}{3}

$x = \frac{\frac{π}{2}}{3}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Multiply the numerator by the reciprocal of the denominator.

x = \frac{π}{2} \cdot \frac{1}{3}

$x = \frac{π}{2} \cdot \frac{1}{3}$

Step 1.4.3.2

Multiply

\frac{π}{2} \cdot \frac{1}{3}

$\frac{π}{2} \cdot \frac{1}{3}$ .

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Step 1.4.3.2.1

Multiply

\frac{π}{2}

$\frac{π}{2}$ by

\frac{1}{3}

$\frac{1}{3}$ .

x = \frac{π}{2 \cdot 3}

$x = \frac{π}{2 \cdot 3}$

Step 1.4.3.2.2

Multiply

2

$2$ by

3

$3$ .

x = \frac{π}{6}

$x = \frac{π}{6}$

x = \frac{π}{6}

$x = \frac{π}{6}$

x = \frac{π}{6}

$x = \frac{π}{6}$

x = \frac{π}{6}

$x = \frac{π}{6}$

Step 1.5

The basic period for

y = \tan (3 x)

$y = \tan (3 x)$ will occur at

(- \frac{π}{6}, \frac{π}{6})

$(- \frac{π}{6}, \frac{π}{6})$ , where

- \frac{π}{6}

$- \frac{π}{6}$ and

\frac{π}{6}

$\frac{π}{6}$ are vertical asymptotes.

(- \frac{π}{6}, \frac{π}{6})

$(- \frac{π}{6}, \frac{π}{6})$

Step 1.6

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

0

$0$ and

3

$3$ is

3

$3$ .

\frac{π}{3}

$\frac{π}{3}$

Step 1.7

The vertical asymptotes for

y = \tan (3 x)

$y = \tan (3 x)$ occur at

- \frac{π}{6}

$- \frac{π}{6}$ ,

\frac{π}{6}

$\frac{π}{6}$ , and every

\frac{π n}{3}

$\frac{π n}{3}$ , where

n

$n$ is an integer.

x = \frac{π}{6} + \frac{π n}{3}

$x = \frac{π}{6} + \frac{π n}{3}$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π}{6} + \frac{π n}{3}

$x = \frac{π}{6} + \frac{π n}{3}$ where

n

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π}{6} + \frac{π n}{3}

$x = \frac{π}{6} + \frac{π n}{3}$ where

n

$n$ is an integer

Step 2

Use the form

a \tan (b x - c) + d

$a \tan (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = 1

$a = 1$

b = 3

$b = 3$

c = 0

$c = 0$

d = 0

$d = 0$

Step 3

Since the graph of the function

t a n

$t a n$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period of

\tan (3 x)

$\tan (3 x)$ .

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Step 4.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 4.2

Replace

b

$b$ with

3

$3$ in the formula for period.

\frac{π}{| 3 |}

$\frac{π}{| 3 |}$

Step 4.3

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

0

$0$ and

3

$3$ is

3

$3$ .

\frac{π}{3}

$\frac{π}{3}$

\frac{π}{3}

$\frac{π}{3}$

Step 5

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 5.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{0}{3}

$\frac{0}{3}$

Step 5.3

Divide

0

$0$ by

3

$3$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

\frac{π}{3}

$\frac{π}{3}$

Phase Shift: None

Vertical Shift: None

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes:

x = \frac{π}{6} + \frac{π n}{3}

$x = \frac{π}{6} + \frac{π n}{3}$ where

n

$n$ is an integer

Amplitude: None

Period:

\frac{π}{3}

$\frac{π}{3}$

Phase Shift: None

Vertical Shift: None

Step 8