Trigonometry Examples

$y = \tan (x + \frac{π}{2})$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \tan (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = \tan (x)$ , $(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for $y = \tan (x + \frac{π}{2})$ . Set the inside of the tangent function, $b x + c$ , for $y = a \tan (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = \tan (x + \frac{π}{2})$ .

$x + \frac{π}{2} = - \frac{π}{2}$

Step 1.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = - \frac{π}{2} - \frac{π}{2}$

Step 1.2.2

Combine the numerators over the common denominator.

$x = \frac{- π - π}{2}$

Step 1.2.3

Subtract $π$ from $- π$ .

$x = \frac{- 2 π}{2}$

Step 1.2.4

Cancel the common factor of $- 2$ and $2$ .

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Step 1.2.4.1

Factor $2$ out of $- 2 π$ .

$x = \frac{2 (- π)}{2}$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- π)}{2 (1)}$

Step 1.2.4.2.2

Cancel the common factor.

$x = \frac{2 (- π)}{2 \cdot 1}$

Step 1.2.4.2.3

Rewrite the expression.

$x = \frac{- π}{1}$

Step 1.2.4.2.4

Divide $- π$ by $1$ .

$x = - π$

Step 1.3

Set the inside of the tangent function $x + \frac{π}{2}$ equal to $\frac{π}{2}$ .

$x + \frac{π}{2} = \frac{π}{2}$

Step 1.4

Move all terms not containing $x$ to the right side of the equation.

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Step 1.4.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = \frac{π}{2} - \frac{π}{2}$

Step 1.4.2

Combine the numerators over the common denominator.

$x = \frac{π - π}{2}$

Step 1.4.3

Subtract $π$ from $π$ .

$x = \frac{0}{2}$

Step 1.4.4

Divide $0$ by $2$ .

$x = 0$

Step 1.5

The basic period for $y = \tan (x + \frac{π}{2})$ will occur at $(- π, 0)$ , where $- π$ and $0$ are vertical asymptotes.

$(- π, 0)$

Step 1.6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 1.6.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 1.6.2

Divide $π$ by $1$ .

$π$

Step 1.7

The vertical asymptotes for $y = \tan (x + \frac{π}{2})$ occur at $- π$ , $0$ , and every $x = - π + π n$ , where $n$ is an integer.

$x = - π + π n$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - π + π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - π + π n$ where $n$ is an integer

Step 2

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

$a = 1$

$b = 1$

$c = - \frac{π}{2}$

$d = 0$

Step 3

Since the graph of the function $t a n$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period of $\tan (x + \frac{π}{2})$ .

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Step 4.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 4.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 4.4

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: $\frac{- \frac{π}{2}}{1}$

Step 5.3

Divide $- \frac{π}{2}$ by $1$ .

Phase Shift: $- \frac{π}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $- \frac{π}{2}$ ( $\frac{π}{2}$ 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: $x = - π + π n$ where $n$ is an integer

Amplitude: None

Period: $π$

Phase Shift: $- \frac{π}{2}$ ( $\frac{π}{2}$ to the left)

Vertical Shift: None

Step 8