Precalculus Examples

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

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

$\frac{x}{2} - \frac{π}{3} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $\frac{π}{3}$ to both sides of the equation.

$\frac{x}{2} = \frac{π}{3}$

Step 2.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{3}$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{3}$

Step 2.3.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Combine $2$ and $\frac{π}{3}$ .

$x = \frac{2 π}{3}$

Step 3

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

$\frac{x}{2} - \frac{π}{3} = 2 π$

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 $\frac{π}{3}$ to both sides of the equation.

$\frac{x}{2} = 2 π + \frac{π}{3}$

Step 4.1.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.1.3

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $3$ by $2$ .

$\frac{x}{2} = \frac{6 π + π}{3}$

Step 4.1.5.2

Add $6 π$ and $π$ .

$\frac{x}{2} = \frac{7 π}{3}$

Step 4.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{7 π}{3}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of $2$ .

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Step 4.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{7 π}{3}$

Step 4.3.1.1.2

Rewrite the expression.

$x = 2 \frac{7 π}{3}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Multiply $2 \frac{7 π}{3}$ .

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Step 4.3.2.1.1

Combine $2$ and $\frac{7 π}{3}$ .

$x = \frac{2 (7 π)}{3}$

Step 4.3.2.1.2

Multiply $7$ by $2$ .

$x = \frac{14 π}{3}$

Step 5

The basic period for $y = \csc (\frac{x}{2} - \frac{π}{3})$ will occur at $(\frac{2 π}{3}, \frac{14 π}{3})$ , where $\frac{2 π}{3}$ and $\frac{14 π}{3}$ are vertical asymptotes.

$(\frac{2 π}{3}, \frac{14 π}{3})$

Step 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 6.1

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 6.3

Multiply $2$ by $2$ .

$4 π$

Step 7

The vertical asymptotes for $y = \csc (\frac{x}{2} - \frac{π}{3})$ occur at $\frac{2 π}{3}$ , $\frac{14 π}{3}$ , and every $x = \frac{2 π}{3} + 2 π n$ , where $n$ is an integer. This is half of the period.

$x = \frac{2 π}{3} + 2 π n$

Step 8

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{2 π}{3} + 2 π n$ where $n$ is an integer

Step 9