Trigonometry Examples

y = \csc (\frac{x}{5})

$y = \csc (\frac{x}{5})$

Step 1

Find the asymptotes.

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Step 1.1

For any

y = \csc (x)

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

x = n π

$x = n π$ , where

n

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

y = c s c (x)

$y = c s c (x)$ ,

(0, 2 π)

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

y = \csc (\frac{x}{5})

$y = \csc (\frac{x}{5})$ . Set the inside of the cosecant function,

b x + c

$b x + c$ , for

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

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

0

$0$ to find where the vertical asymptote occurs for

y = \csc (\frac{x}{5})

$y = \csc (\frac{x}{5})$ .

\frac{x}{5} = 0

$\frac{x}{5} = 0$

Step 1.2

Set the numerator equal to zero.

x = 0

$x = 0$

Step 1.3

Set the inside of the cosecant function

\frac{x}{5}

$\frac{x}{5}$ equal to

2 π

$2 π$ .

\frac{x}{5} = 2 π

$\frac{x}{5} = 2 π$

Step 1.4

Solve for

x

$x$ .

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Step 1.4.1

Multiply both sides of the equation by

5

$5$ .

5 \frac{x}{5} = 5 (2 π)

$5 \frac{x}{5} = 5 (2 π)$

Step 1.4.2

Simplify both sides of the equation.

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Step 1.4.2.1

Simplify the left side.

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Step 1.4.2.1.1

Cancel the common factor of

5

$5$ .

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Step 1.4.2.1.1.1

Cancel the common factor.

5 \frac{x}{5} = 5 (2 π)

$5 \frac{x}{5} = 5 (2 π)$

Step 1.4.2.1.1.2

Rewrite the expression.

x = 5 (2 π)

$x = 5 (2 π)$

x = 5 (2 π)

$x = 5 (2 π)$

x = 5 (2 π)

$x = 5 (2 π)$

Step 1.4.2.2

Simplify the right side.

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Step 1.4.2.2.1

Multiply

2

$2$ by

5

$5$ .

x = 10 π

$x = 10 π$

x = 10 π

$x = 10 π$

x = 10 π

$x = 10 π$

x = 10 π

$x = 10 π$

Step 1.5

The basic period for

y = \csc (\frac{x}{5})

$y = \csc (\frac{x}{5})$ will occur at

(0, 10 π)

$(0, 10 π)$ , where

0

$0$ and

10 π

$10 π$ are vertical asymptotes.

(0, 10 π)

$(0, 10 π)$

Step 1.6

Find the period

\frac{2 π}{| b |}

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

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Step 1.6.1

\frac{1}{5}

$\frac{1}{5}$ is approximately

0.2

$0.2$ which is positive so remove the absolute value

\frac{2 π}{\frac{1}{5}}

$\frac{2 π}{\frac{1}{5}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

2 π \cdot 5

$2 π \cdot 5$

Step 1.6.3

Multiply

5

$5$ by

2

$2$ .

10 π

$10 π$

10 π

$10 π$

Step 1.7

The vertical asymptotes for

y = \csc (\frac{x}{5})

$y = \csc (\frac{x}{5})$ occur at

0

$0$ ,

10 π

$10 π$ , and every

5 π n

$5 π n$ , where

n

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

x = 5 π n

$x = 5 π n$

Step 1.8

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = 5 π n

$x = 5 π n$ where

n

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = 5 π n

$x = 5 π n$ where

n

$n$ is an integer

Step 2

Use the form

a \csc (b x - c) + d

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

a = 1

$a = 1$

b = \frac{1}{5}

$b = \frac{1}{5}$

c = 0

$c = 0$

d = 0

$d = 0$

Step 3

Since the graph of the function

c s c

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

Amplitude: None

Step 4

Find the period of

\csc (\frac{x}{5})

$\csc (\frac{x}{5})$ .

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Step 4.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 4.2

Replace

b

$b$ with

\frac{1}{5}

$\frac{1}{5}$ in the formula for period.

\frac{2 π}{| \frac{1}{5} |}

$\frac{2 π}{| \frac{1}{5} |}$

Step 4.3

\frac{1}{5}

$\frac{1}{5}$ is approximately

0.2

$0.2$ which is positive so remove the absolute value

\frac{2 π}{\frac{1}{5}}

$\frac{2 π}{\frac{1}{5}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

2 π \cdot 5

$2 π \cdot 5$

Step 4.5

Multiply

5

$5$ by

2

$2$ .

10 π

$10 π$

10 π

$10 π$

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}{\frac{1}{5}}

$\frac{0}{\frac{1}{5}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

0 \cdot 5

$0 \cdot 5$

Step 5.4

Multiply

0

$0$ by

5

$5$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

10 π

$10 π$

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 = 5 π n

$x = 5 π n$ where

n

$n$ is an integer

Amplitude: None

Period:

10 π

$10 π$

Phase Shift: None

Vertical Shift: None

Step 8