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: $- \frac{π}{8}$

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