Trigonometry Examples

$y = - 2 \csc (2 x + \frac{π}{4})$

Step 1

Find the asymptotes.

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Step 1.1

For any

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

$x = n π$ , where

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

$y = c s c (x)$ ,

$(0, 2 π)$ , to find the vertical asymptotes for

$y = - 2 \csc (2 x + \frac{π}{4})$ . Set the inside of the cosecant function,

$b x + c$ , for

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

$0$ to find where the vertical asymptote occurs for

$y = - 2 \csc (2 x + \frac{π}{4})$ .

$2 x + \frac{π}{4} = 0$

Step 1.2

Solve for

$x$ .

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Step 1.2.1

Subtract

$\frac{π}{4}$ from both sides of the equation.

$2 x = - \frac{π}{4}$

Step 1.2.2

Divide each term in

$2 x = - \frac{π}{4}$ by

$2$ and simplify.

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Step 1.2.2.1

Divide each term in

$2 x = - \frac{π}{4}$ by

$2$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of

$2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- \frac{π}{4}}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{2}$

Step 1.2.2.3.2

Multiply

$- \frac{π}{4} \cdot \frac{1}{2}$ .

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Step 1.2.2.3.2.1

Multiply

$\frac{1}{2}$ by

$\frac{π}{4}$ .

$x = - \frac{π}{2 \cdot 4}$

Step 1.2.2.3.2.2

Multiply

$2$ by

$4$ .

$x = - \frac{π}{8}$

Step 1.3

Set the inside of the cosecant function

$2 x + \frac{π}{4}$ equal to

$2 π$ .

$2 x + \frac{π}{4} = 2 π$

Step 1.4

Solve for

$x$ .

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Step 1.4.1

Move all terms not containing

$x$ to the right side of the equation.

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Step 1.4.1.1

Subtract

$\frac{π}{4}$ from both sides of the equation.

$2 x = 2 π - \frac{π}{4}$

Step 1.4.1.2

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 1.4.1.3

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 1.4.1.4

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 4 - π}{4}$

Step 1.4.1.5

Simplify the numerator.

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Step 1.4.1.5.1

Multiply

$4$ by

$2$ .

$2 x = \frac{8 π - π}{4}$

Step 1.4.1.5.2

Subtract

$π$ from

$8 π$ .

$2 x = \frac{7 π}{4}$

Step 1.4.2

Divide each term in

$2 x = \frac{7 π}{4}$ by

$2$ and simplify.

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Step 1.4.2.1

Divide each term in

$2 x = \frac{7 π}{4}$ by

$2$ .

$\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of

$2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2}$

Step 1.4.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{7 π}{4}}{2}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{4} \cdot \frac{1}{2}$

Step 1.4.2.3.2

Multiply

$\frac{7 π}{4} \cdot \frac{1}{2}$ .

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Step 1.4.2.3.2.1

Multiply

$\frac{7 π}{4}$ by

$\frac{1}{2}$ .

$x = \frac{7 π}{4 \cdot 2}$

Step 1.4.2.3.2.2

Multiply

$4$ by

$2$ .

$x = \frac{7 π}{8}$

Step 1.5

The basic period for

$y = - 2 \csc (2 x + \frac{π}{4})$ will occur at

$(- \frac{π}{8}, \frac{7 π}{8})$ , where

$- \frac{π}{8}$ and

$\frac{7 π}{8}$ are vertical asymptotes.

$(- \frac{π}{8}, \frac{7 π}{8})$

Step 1.6

Find the period

$\frac{2 π}{| b |}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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Step 1.6.1

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

$0$ and

$2$ is

$2$ .

$\frac{2 π}{2}$

Step 1.6.2

Cancel the common factor of

$2$ .

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Step 1.6.2.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.6.2.2

Divide

$π$ by

$1$ .

$π$

Step 1.7

The vertical asymptotes for

$y = - 2 \csc (2 x + \frac{π}{4})$ occur at

$- \frac{π}{8}$ ,

$\frac{7 π}{8}$ , and every

$x = - \frac{π}{8} + \frac{π n}{2}$ , where

$n$ is an integer. This is half of the period.

$x = - \frac{π}{8} + \frac{π n}{2}$

Step 1.8

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = - \frac{π}{8} + \frac{π n}{2}$ where

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = - \frac{π}{8} + \frac{π n}{2}$ where

$n$ is an integer

Step 2

Use the form

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

$a = - 2$

$b = 2$

$c = - \frac{π}{4}$

$d = 0$

Step 3

Since the graph of the function

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

Amplitude: None

Step 4

Find the period of

$- 2 \csc (2 x + \frac{π}{4})$ .

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Step 4.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2

Replace

$b$ with

$2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 4.3

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

$0$ and

$2$ is

$2$ .

$\frac{2 π}{2}$

Step 4.4

Cancel the common factor of

$2$ .

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Step 4.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.4.2

Divide

$π$ by

$1$ .

$π$

Step 5

Find the phase shift using the formula

$\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from

$\frac{c}{b}$ .

Phase Shift:

$\frac{c}{b}$

Step 5.2

Replace the values of

$c$ and

$b$ in the equation for phase shift.

Phase Shift:

$\frac{- \frac{π}{4}}{2}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$- \frac{π}{4} \cdot \frac{1}{2}$

Step 5.4

Multiply

$- \frac{π}{4} \cdot \frac{1}{2}$ .

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Step 5.4.1

Multiply

$\frac{1}{2}$ by

$\frac{π}{4}$ .

Phase Shift:

$- \frac{π}{2 \cdot 4}$

Step 5.4.2

Multiply

$2$ by

$4$ .

Phase Shift:

$- \frac{π}{8}$

Phase Shift:

$- \frac{π}{8}$

Phase Shift:

$- \frac{π}{8}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

$π$

Phase Shift:

$- \frac{π}{8}$ (

$\frac{π}{8}$ to the left)

Vertical Shift: None

Step 7

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

Vertical Asymptotes:

$x = - \frac{π}{8} + \frac{π n}{2}$ where

$n$ is an integer

Amplitude: None

Period:

$π$

Phase Shift:

$- \frac{π}{8}$ (

$\frac{π}{8}$ to the left)

Vertical Shift: None

Step 8