Trigonometry Examples

y = \cot (2 x)

$y = \cot (2 x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

For any

y = \cot (x)

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

x = n π

$x = n π$ , where

n

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

y = \cot (x)

$y = \cot (x)$ ,

(0, π)

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

y = \cot (2 x)

$y = \cot (2 x)$ . Set the inside of the cotangent function,

b x + c

$b x + c$ , for

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

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

0

$0$ to find where the vertical asymptote occurs for

y = \cot (2 x)

$y = \cot (2 x)$ .

2 x = 0

$2 x = 0$

Step 1.2

Divide each term in

2 x = 0

$2 x = 0$ by

2

$2$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in

2 x = 0

$2 x = 0$ by

2

$2$ .

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

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

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

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

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

Step 1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{2}

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

x = \frac{0}{2}

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

x = \frac{0}{2}

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

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Divide

0

$0$ by

2

$2$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 1.3

Set the inside of the cotangent function

2 x

$2 x$ equal to

π

$π$ .

2 x = π

$2 x = π$

Step 1.4

Divide each term in

2 x = π

$2 x = π$ by

2

$2$ and simplify.

Tap for more steps...

Step 1.4.1

Divide each term in

2 x = π

$2 x = π$ by

2

$2$ .

\frac{2 x}{2} = \frac{π}{2}

$\frac{2 x}{2} = \frac{π}{2}$

Step 1.4.2

Simplify the left side.

Tap for more steps...

Step 1.4.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 1.4.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{π}{2}

$\frac{2 x}{2} = \frac{π}{2}$

Step 1.4.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{π}{2}

$x = \frac{π}{2}$

x = \frac{π}{2}

$x = \frac{π}{2}$

x = \frac{π}{2}

$x = \frac{π}{2}$

x = \frac{π}{2}

$x = \frac{π}{2}$

Step 1.5

The basic period for

y = \cot (2 x)

$y = \cot (2 x)$ will occur at

(0, \frac{π}{2})

$(0, \frac{π}{2})$ , where

0

$0$ and

\frac{π}{2}

$\frac{π}{2}$ are vertical asymptotes.

(0, \frac{π}{2})

$(0, \frac{π}{2})$

Step 1.6

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

0

$0$ and

2

$2$ is

2

$2$ .

\frac{π}{2}

$\frac{π}{2}$

Step 1.7

The vertical asymptotes for

y = \cot (2 x)

$y = \cot (2 x)$ occur at

0

$0$ ,

\frac{π}{2}

$\frac{π}{2}$ , and every

\frac{π n}{2}

$\frac{π n}{2}$ , where

n

$n$ is an integer.

x = \frac{π n}{2}

$x = \frac{π n}{2}$

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π n}{2}

$x = \frac{π n}{2}$ where

n

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π n}{2}

$x = \frac{π n}{2}$ where

n

$n$ is an integer

Step 2

Use the form

a \cot (b x - c) + d

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

a = 1

$a = 1$

b = 2

$b = 2$

c = 0

$c = 0$

d = 0

$d = 0$

Step 3

Since the graph of the function

c o t

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

Amplitude: None

Step 4

Find the period of

\cot (2 x)

$\cot (2 x)$ .

Tap for more steps...

Step 4.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 4.2

Replace

b

$b$ with

2

$2$ in the formula for period.

\frac{π}{| 2 |}

$\frac{π}{| 2 |}$

Step 4.3

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

0

$0$ and

2

$2$ is

2

$2$ .

\frac{π}{2}

$\frac{π}{2}$

\frac{π}{2}

$\frac{π}{2}$

Step 5

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

Tap for more steps...

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}{2}

$\frac{0}{2}$

Step 5.3

Divide

0

$0$ by

2

$2$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

\frac{π}{2}

$\frac{π}{2}$

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 = \frac{π n}{2}

$x = \frac{π n}{2}$ where

n

$n$ is an integer

Amplitude: None

Period:

\frac{π}{2}

$\frac{π}{2}$

Phase Shift: None

Vertical Shift: None

Step 8