Trigonometry Examples

y = tan (2 x - π)

$y = \tan (2 x - π)$

Step 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 (2 x - π)

$y = \tan (2 x - π)$ . 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 (2 x - π)

$y = \tan (2 x - π)$ .

2 x - π = - \frac{π}{2}

$2 x - π = - \frac{π}{2}$

Step 2

Solve for

x

$x$ .

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Step 2.1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1.1

Add

π

$π$ to both sides of the equation.

2 x = - \frac{π}{2} + π

$2 x = - \frac{π}{2} + π$

Step 2.1.2

To write

π

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

\frac{2}{2}

$\frac{2}{2}$ .

2 x = - \frac{π}{2} + π \cdot \frac{2}{2}

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

Step 2.1.3

Combine

π

$π$ and

\frac{2}{2}

$\frac{2}{2}$ .

2 x = - \frac{π}{2} + \frac{π \cdot 2}{2}

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

Step 2.1.4

Combine the numerators over the common denominator.

2 x = \frac{- π + π \cdot 2}{2}

$2 x = \frac{- π + π \cdot 2}{2}$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Move

2

$2$ to the left of

π

$π$ .

2 x = \frac{- π + 2 \cdot π}{2}

$2 x = \frac{- π + 2 \cdot π}{2}$

Step 2.1.5.2

Add

- π

$- π$ and

2 π

$2 π$ .

2 x = \frac{π}{2}

$2 x = \frac{π}{2}$

2 x = \frac{π}{2}

$2 x = \frac{π}{2}$

2 x = \frac{π}{2}

$2 x = \frac{π}{2}$

Step 2.2

Divide each term in

2 x = \frac{π}{2}

$2 x = \frac{π}{2}$ by

2

$2$ and simplify.

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Step 2.2.1

Divide each term in

2 x = \frac{π}{2}

$2 x = \frac{π}{2}$ by

2

$2$ .

\frac{2 x}{2} = \frac{\frac{π}{2}}{2}

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.2.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.2.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.2.3.2.2

Multiply

$2$ by

$2$ .

$x = \frac{π}{4}$

Step 3

Set the inside of the tangent function

$2 x - π$ equal to

$\frac{π}{2}$ .

$2 x - π = \frac{π}{2}$

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

$π$ to both sides of the equation.

$2 x = \frac{π}{2} + π$

Step 4.1.2

To write

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

$\frac{2}{2}$ .

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

Step 4.1.3

Combine

$π$ and

$\frac{2}{2}$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

$2 x = \frac{π + π \cdot 2}{2}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Move

$2$ to the left of

$π$ .

$2 x = \frac{π + 2 \cdot π}{2}$

Step 4.1.5.2

Add

$π$ and

$2 π$ .

$2 x = \frac{3 π}{2}$

Step 4.2

Divide each term in

$2 x = \frac{3 π}{2}$ by

$2$ and simplify.

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Step 4.2.1

Divide each term in

$2 x = \frac{3 π}{2}$ by

$2$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of

$2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 4.2.3.2

Multiply

$\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 4.2.3.2.1

Multiply

$\frac{3 π}{2}$ by

$\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 4.2.3.2.2

Multiply

$2$ by

$2$ .

$x = \frac{3 π}{4}$

Step 5

The basic period for

$y = \tan (2 x - π)$ will occur at

$(\frac{π}{4}, \frac{3 π}{4})$ , where

$\frac{π}{4}$ and

$\frac{3 π}{4}$ are vertical asymptotes.

$(\frac{π}{4}, \frac{3 π}{4})$

Step 6

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

$0$ and

$2$ is

$2$ .

$\frac{π}{2}$

Step 7

The vertical asymptotes for

$y = \tan (2 x - π)$ occur at

$\frac{π}{4}$ ,

$\frac{3 π}{4}$ , and every

$x = \frac{π}{4} + \frac{π n}{2}$ , where

$n$ is an integer.

$x = \frac{π}{4} + \frac{π n}{2}$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = \frac{π}{4} + \frac{π n}{2}$ where

$n$ is an integer

Step 9