Graphs of Other Trigonometric Functions


Function	Domain

$\tan θ$	All real numbers except odd multiples of $\frac{π}{2}$ ( $90^{\circ}$ )

$\cot θ$	All real numbers except integer multiples of $π$ ( $180^{\circ}$ )

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Subsection 2.2.2 Ranges of the Tangent and Cotangent Functions

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To determine the range of the tangent function, consider the point

P (x, y)

on the unit circle corresponding to the angle

θ,

and let

a

be a real number such that

a = \tan θ = \frac{y}{x} .

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Multiplying both sides by

x,

we obtain:

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y = a x

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Squaring both sides yields:

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y^{2} = a^{2} x^{2}

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Substituting into the Pythagorean Identity (Definition 1.5.20), we have:

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1 = x^{2} + y^{2} = x^{2} + a^{2} x^{2} = x^{2} (1 + a^{2})

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Dividing both sides by

1 + a^{2}

and taking the square root gives:

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x = \pm \frac{1}{\sqrt{1 + a^{2}}}

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Similarly, we obtain:

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y = \pm \frac{a}{\sqrt{1 + a^{2}}}

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Thus, we conclude:

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\tan θ = \frac{y}{x} = \frac{\frac{a}{\sqrt{1 + a^{2}}}}{\frac{1}{\sqrt{1 + a^{2}}}} = a

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In other words, since

a

can be any real number and

\tan θ = a,

the range of the tangent function consists of all real numbers. A similar method can be used to show that the range of the cotangent function is also the set of all real numbers.

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Remark 2.2.3. Ranges for Tangent and Cotangent.

The ranges for tangent and cotangent functions are given in Table 2.2.4

Table 2.2.4. Ranges of the tangent and cotangent functions


Function	Range

$\tan θ$	All real numbers

$\cot θ$	All real numbers

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Subsection 2.2.3 Domains of the Cosecant and Secant Functions

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Consider the Reciprocal Identity for the cosecant function (Definition 1.4.2):

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\csc x = \frac{1}{\sin x}

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When

\sin x = 0,

corresponding to

x = \dots, - π, 0, π, 2 π, \dots,

the denominator becomes zero. In general,

\csc x

is undefined for angles of the form

n π,

where

n

is an integer, and these values should be excluded from the domain.

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Similarly, since the cosecant function is defined as

\sec x = \frac{1}{\cos x}

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we see that

\sec x

is undefined when

\cos x = 0 .

This occurs at

x = \dots, - \frac{3 π}{2}, - \frac{π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \dots,

and thus any angle of the general form

n \frac{π}{2},

where

n

is an odd integer, should be excluded from the domain of

\sec x .

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Remark 2.2.5. Domains for Cosecant and Secant.

The domains for cosecant and secant functions are given in Table 2.2.6

Table 2.2.6. Domains of the trigonometric functions


Function	Domain

$\csc θ$	All real numbers except integer multiples of $π$ ( $180^{\circ}$ )

$\sec θ$	All real numbers except integer multiples of $\frac{π}{2}$ ( $90^{\circ}$ )

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Subsection 2.2.4 Ranges of the Cosecant and Secant Functions

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If the angle is not an integer multiple of

π,

i.e.

x \neq n π,

where

n

is an integer, then the Reciprocal Identity allows us to define cosecant as

\csc x = \frac{1}{\sin x} .

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In Subsection 2.1.1, we learned that the function

y = \sin x

has a range of

- 1 \leq \sin x \leq 1 .

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Therefore, taking the reciprocal of the range of sine, we get

\csc x \leq - 1 or \csc x \geq 1 .

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In other words, the range of the cosecant function is all real numbers less than or equal to

- 1

or greater than or equal to

1 .

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Similarly, since

- 1 \leq \cos x \leq 1,

we can get the range for the secant funcation as

\sec x \leq - 1 or \sec x \geq 1

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or all real numbers less than or equal to

- 1

or greater than or equal to

1 .

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Remark 2.2.7. Ranges for Cosecant and Secant.

The ranges for cosecant and secant functions are given in Table 2.2.8

Table 2.2.8. Ranges of the cosecant and secant functions


Function	Range

$\csc θ$	All real numbers greater than or equal to $1$ or less than or equal to $- 1$

$\sec θ$	All real numbers greater than or equal to $1$ or less than or equal to $- 1$

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Subsection 2.2.5 The Tangent Function

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In Subsection 1.5.4, we learned that the tangent function is periodic with a period of

π .

To graph

y = \tan x,

we focus on plotting one period and then repeat those values to complete the graph.

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We also know that the domain of tangent includes all real numbers except any angle of the form

n \frac{π}{2},

where

n

is an odd integer. These values are excluded since tangent is undefined there. In fact, any line of the form

x = n \frac{π}{2}

(e.g.

x = - \frac{π}{2}

and

x = \frac{π}{2}

) is a vertical asymptote

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Knowing the location of the vertical asymptotes, we choose the interval

(- \frac{π}{2}, \frac{π}{2})

to plot our points for tangent. This interval has a length of

π

(one period), allowing us to repeat the values to complete the graph of tangent over the entire domain.

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Recall from Definition 1.3.2 that on the unit circle, the expression

\tan θ = \frac{y}{x}

denotes the ratio of the

y

-coordinate to the

x

-coordinate of a point

P (x, y)

associated with an angle

θ .

This ratio changes dynamically as

θ

varies from zero to

\frac{π}{2} .

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θ

approaches zero, the

y

-value tends to zero, and

x

approaches 1, resulting in

\tan θ

being a small fraction. Conversely, as

θ

approaches

\frac{π}{2},

the

y

-value approaches 1, while

x

becomes extremely small and near zero. This causes

\tan θ

to evaluate as a fraction divided by a very small number, producing a large number. A similar effect occurs when

θ

is between

- \frac{π}{2}

and zero, except

\tan θ

is negative in this range. This behavior is shown in Figure 2.2.9.

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Figure 2.2.9. As

θ

moves from

- 90^{\circ}

90^{\circ},

this figure plots the values of

y = \tan θ .

Move the slider for

θ

to see how changing the angle affects

\tan θ .

Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.

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Next we recall values of tangent for known angles, which are listed in Table 2.2.10 and plotted in Figure 2.2.11.

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Table 2.2.10. Values for

y = \tan x

$x$	$\tan x$	$(x, y)$
$- \frac{π}{3}$	$- \sqrt{3}$	$(- \frac{π}{3}, - \sqrt{3})$
$- \frac{π}{4}$	$- 1$	$(- \frac{π}{4}, - 1)$
$- \frac{π}{6}$	$- \frac{\sqrt{3}}{3}$	$(- \frac{π}{6}, - \frac{\sqrt{3}}{3})$
$0$	$0$	$(0, 0)$
$\frac{π}{6}$	$\frac{\sqrt{3}}{3}$	$(\frac{π}{6}, \frac{\sqrt{3}}{3})$
$\frac{π}{4}$	$1$	$(\frac{π}{4}, 1)$
$\frac{π}{3}$	$\sqrt{3}$	$(\frac{π}{3}, \sqrt{3})$

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Figure 2.2.11. The values in Table 2.2.10 plotted on a graph with a smooth curve connecting the points to make the curve for $y = \tan (x) .$

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Notice the symmetrical nature of the tangent function’s graph with respect to the origin, a feature explained in Section 1.5.7, where we learned that tangent is an odd function.

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Since the graph in Figure 2.2.11 represents one period, we can complete the graph of

y = \tan x

by extending the pattern in both directions to obtain Figure 2.2.12.

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Figure 2.2.12. The plot for $y = \tan (x)$

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Subsection 2.2.6 The Cotangent Function

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The domain of the cotangent function is defined for all angles except those of the form

n π,

where

n

is an integer. These excluded values correspond to vertical asymptotes. In fact, any line of the form

x = n π,

where

n

is an integer, serves as a vertical asymptote. Additionally, as we learned in Subsection 1.5.4, the cotangent function has a period of

π .

We can construct a plot for it in a manner similar to how we constructed the tangent function, as illustrated in Figure 2.2.13.

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Figure 2.2.13. As

θ

moves from

0^{\circ}

180^{\circ},

this figure plots the values of

y = \cot θ .

Move the slider for

θ

to see how changing the angle affects

\cot θ .

Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.

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Plotting points for

y = \cot x

and using the fact that the cotangent function is periodic, we obtain the graph for cotangent in Figure 2.2.14

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Figure 2.2.14. The plot for $y = \cot (x) .$

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The symmetry of the cotangent function about the origin is evident, confirming its nature as an odd function, as explained in Section 1.5.7.

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Subsection 2.2.7 The Cosecant Function

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The cosecant function has vertical asymptotes at points

n π,

where

n

is an integer, corresponding to the values where the function is undefined. These points are the same ones excluded from the domain of

\csc x .

As discussed in Subsection 1.5.4, the cosecant function has a period of

2 π .

Given this, we choose to examine its behavior within one period, specifically from

0

2 π,

since this interval spans one complete period of the cosecant function.

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Consider the Reciprocal Identity for the cosecant function (Definition 1.4.2):

\csc x = \frac{1}{\sin x} .

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x

approaches zero, sine decreases to zero, making cosecant approach positive infinity. Increasing

x

towards

\frac{π}{2},

\sin (x)

increases to

1,

and cosecant decreases to

1 .

x

moves from

\frac{π}{2}

π,

sine approaches zero, causing

\csc (x)

to approach infinity.

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Similarly, for

x > π

nearing

π,

\sin (x)

becomes a small, negative number near zero, resulting in

\csc (x)

approaching negative infinity. As

x

increases to

\frac{3 π}{2},

\sin (x)

decreases to

- 1,

and

\csc (x)

increases to

- 1 .

Finally, from

\frac{3 π}{2}

2 π,

\sin (x)

approaches a small negative number, causing cosecant to approach negative infinity. This behavior is shown in Figure 2.2.15.

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Figure 2.2.15. As

θ

moves from

0^{\circ}

360^{\circ},

this figure plots the values of

y = \sin x

and

y = \csc θ .

Move the slider for

θ

to see how changing the angle affects

\csc θ .

Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.

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Next we recall values of sine and cosecant for known angles, which are listed in Table 2.2.16 and plotted in Figure 2.2.17.

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Table 2.2.16. Values for

y = \csc x

$x$	$\sin x$	$\csc x$	$x$	$\sin x$	$\csc x$
$0$	$0$	Undefined	$π$	$0$	Undefined
$\frac{π}{6}$	$\frac{1}{2}$	$2$	$\frac{7 π}{6}$	$- \frac{1}{2}$	$- 2$
$\frac{π}{4}$	$\frac{\sqrt{2}}{2}$	$\sqrt{2}$	$\frac{5 π}{4}$	$- \frac{\sqrt{2}}{2}$	$- \sqrt{2}$
$\frac{π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{2 \sqrt{3}}{3}$	$\frac{4 π}{3}$	$- \frac{\sqrt{3}}{2}$	$- \frac{2 \sqrt{3}}{3}$
$\frac{π}{2}$	$1$	$1$	$\frac{3 π}{2}$	$- 1$	$- 1$
$\frac{2 π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{2 \sqrt{3}}{3}$	$\frac{5 π}{3}$	$- \frac{\sqrt{3}}{2}$	$- \frac{2 \sqrt{3}}{3}$
$\frac{3 π}{4}$	$\frac{\sqrt{2}}{2}$	$\sqrt{2}$	$\frac{7 π}{4}$	$- \frac{\sqrt{2}}{2}$	$- \sqrt{2}$
$\frac{5 π}{6}$	$\frac{1}{2}$	$2$	$\frac{11 π}{6}$	$- \frac{1}{2}$	$- 2$
			$2 π$	$0$	Undefined

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Figure 2.2.17. The values in Table 2.2.16 plotted on a graph with a smooth curve connecting the points to make the curve for $y = \csc (x) .$

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Since the graph in Figure 2.2.17 represents one period, we can complete the graph of

y = \csc x

by extending the pattern in both directions to obtain Figure 2.2.18.

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Figure 2.2.18. The plot for $y = \sin (x)$ and $y = \csc (x) .$

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Notice the graph of the cosecant function is symmetric with respect to the origin, confirming what we learned in Section 1.5.7, that cosecant is an odd function.

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Subsection 2.2.8 The Secant Function

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As discussed earlier in this section, the secant function has a domain for all real numbers except angles of

n \frac{π}{2},

where

n

is an integer. These excluded values correspond to the vertical asymptotes of the secant function. With a period of

2 π,

we can focus on the interval

0

2 π .

The construction of the secant function plot follows a similar approach to that used for the cosecant function, as illustrated in Figure 2.2.19.

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Figure 2.2.19. As

θ

moves from

0^{\circ}

360^{\circ},

this figure plots the values of

y = \cos x

and

y = \sec θ .

Move the slider for

θ

to see how changing the angle affects

\sec θ .

Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.

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Plotting points for

y = \sec x

and using the fact that the secant function is periodic, we obtain the graph for secant in Figure 2.2.20.

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Figure 2.2.20. The plot for $y = \cos (x)$ and $y = \sec (x) .$

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Notice the graph of the secant function is symmetric about the

y

-axis, and thus secant is an even function, confirming what we learned in Section 1.5.7.

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Subsection 2.2.9 Graphing Transformations of Other Trigonometric Functions

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Similar to the graphs of sine and cosine, the graphs of the other trigonometric functions can undergo vertical stretching and compressing, horizontal stretching and compressing, phase shifts, vertical shift transformations, and reflections about the

x

- and

y

-axes. However, unlike the sine and cosine functions, there is no amplitude for the other trigonometric functions. These transformations are listed in Definition 2.2.21.

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Definition 2.2.21. Transformations of the Tangent, Cotangent, Cosecant, and Secant Functions.

For functions of the form

y = A \cdot \tan (B (x - C)) + D, y = A \cdot \cot (B (x - C)) + D,

y = A \cdot \csc (B (x - C)) + D, and y = A \cdot \sec (B (x - C)) + D,

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we can express the transformations as follows:

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Vertical Compression/Stretch: $| A |$
- $| A |$ is the value of the vertical stretch/compression.
- If $| A | > 1,$ there is vertical stretching.
- If $0 < | A | < 1,$ there is vertical compression.
Period and Horizontal Stretch/Compression: $| B |$
- The period is $\frac{π}{| B |}$ for tangent and cotangent, and $\frac{2 π}{| B |}$ for cosecant and secant.
- If $| B | > 1,$ there is horizontal compression, and the period is shortened.
- If $0 < | B | < 1,$ there is horizontal stretching, and the period is lengthened.
Phase Shift: $C$
- If $C$ is positive, there is a shift to the right.
- If $C$ is negative, there is a shift to the left.
Vertical Shift: $D$
- If $D$ is positive, there is a shift upward.
- If $D$ is negative, there is a shift downward.
Reflection about the $x$ -axis:
- If $A$ is negative ( $A < 0$ ), there is a reflection about the $x$ -axis.
Reflection about the $y$ -axis:
- If $B$ is negative ( $B < 0$ ), there is a reflection about the $y$ -axis.
Vertical Asymptotes:
- For tangent and cotangent, vertical asymptotes occur at
  
  $x = C + \frac{π}{| B |} n,$
  
  where $n$ is an integer.
- For cosecant and secant, vertical asymptotes occur at
  
  $x = C + \frac{π}{2 | B |} n,$
  
  where $n$ is an integer.

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Remark 2.2.22. Other Forms of Transformations.

For functions of the form

y = A \cdot \tan (B x - E) + D, y = A \cdot \cot (B x - E) + D,

y = A \cdot \csc (B x - E) + D, and y = A \cdot \sec (B x - E) + D,

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the transformations are the same as above, except for the phase shift and vertical asymptotes where you replace

C

with

\frac{E}{B} .

\frac{E}{B} < 0

the phase shift is to the right, and if

\frac{E}{B} > 0

it is to the left.

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For tangent and cotangent, vertical asymptotes occur at

$x = \frac{E}{B} + \frac{π}{| B |} n,$

where $n$ is an integer.
For cosecant and secant, vertical asymptotes occur at

$x = \frac{E}{B} + \frac{π}{2 | B |} n,$

where $n$ is an integer.

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Example 2.2.23. Vertical Stretch/Compression and Reflection about the $x$ -axis.

Graph each function

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$y = \tan (x)$
$y = 2 \tan (x)$
$y = \frac{1}{2} \tan (x)$
$y = - \tan (x)$

Solution.

Figure 2.2.24. The transformations of the tangent function graph, starting with a baseline of $y = \tan (x)$ along with graphs with a vertical stretch, a vertical compression, and a reflection about the $x$ -axis.

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Example 2.2.25. Horizontal Stretch/Compression and Reflection about the $y$ -axis.

Identify the period and graph one period for each of the following functions:

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$y = \cot (x)$
$y = \cot (2 x)$
$y = \cot (\frac{1}{2} x)$
$y = \cot (- x)$

Solution.

The period for $y = \cot (x)$ is $π$
The period for $y = \cot (2 x)$ is

$period = \frac{π}{| 2 |} = \frac{π}{2}$
The period for $y = \cot (\frac{1}{2} x)$ is

$period = \frac{π}{| \frac{1}{2} |} = 2 π$
The period for $y = \cot (- x)$ is

$period = \frac{π}{| - 1 |} = π$

Figure 2.2.26. The transformations of the cotangent function graph, starting with a baseline of $y = \cot (x)$ along with graphs with a horizontal stretch, a horizontal compression, and a reflection about the $y$ -axis.

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Example 2.2.27. Phase Shifts and Vertical Shifts.

Graph the function

y = \csc (x - \frac{π}{2}) + 2 .

Solution.

Figure 2.2.28. The graphs of $y = \csc (x - \frac{π}{2}) + 2$ and $y = \sin (x - \frac{π}{2}) + 2$ start with the plots of $y = \csc (x)$ and $y = \sin (x)$ and add a phase shift of $\frac{π}{2}$ to the right and a vertical shift up by $2 .$

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Explore the effects of various transformations using the interactive features in Figure 2.2.29 and Figure 2.2.30.

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Figure 2.2.29. Manipulate the graphs of tangent and cotangent by adjusting the sliders for

A,

B,

C,

and

D .

Observe the effects on period, phase and vertical shifts, as well as reflections about the

x

- and

y

-axes. Additionally, you can toggle between the tangent and cotangent graphs by selecting the corresponding function.

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Figure 2.2.30. Manipulate the graphs of cosecant and secant by adjusting the sliders for

A,

B,

C,

and

D .

Observe the effects on period, phase and vertical shifts, as well as reflections about the

x

- and

y

-axes. Additionally, you can toggle between the cosecant and secant graphs by selecting the corresponding function.

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Exercises 2.2.10 Exercises

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Exercise Group.

Graph the function.

🔗

1.

y = \sec (x) - 4

Section 2.2 Graphs of Other Trigonometric Functions

Subsection 2.2.1 Domains of the Tangent and Cotangent Functions

Remark 2.2.1. Domains for Tangent and Cotangent.

Subsection 2.2.2 Ranges of the Tangent and Cotangent Functions

Remark 2.2.3. Ranges for Tangent and Cotangent.

Subsection 2.2.3 Domains of the Cosecant and Secant Functions

Remark 2.2.5. Domains for Cosecant and Secant.

Subsection 2.2.4 Ranges of the Cosecant and Secant Functions

Remark 2.2.7. Ranges for Cosecant and Secant.

Subsection 2.2.5 The Tangent Function

Subsection 2.2.6 The Cotangent Function

Subsection 2.2.7 The Cosecant Function

Subsection 2.2.8 The Secant Function

Subsection 2.2.9 Graphing Transformations of Other Trigonometric Functions

Definition 2.2.21. Transformations of the Tangent, Cotangent, Cosecant, and Secant Functions.

Remark 2.2.22. Other Forms of Transformations.

Example 2.2.23. Vertical Stretch/Compression and Reflection about the x-axis.

Solution.

Example 2.2.25. Horizontal Stretch/Compression and Reflection about the y-axis.

Solution.

Example 2.2.27. Phase Shifts and Vertical Shifts.

Solution.

Exercises 2.2.10 Exercises

Exercise Group.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Exercise Group.

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Exercise Group.

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23.

24.

25.

26.

Exercise Group.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

Navigating with Shadows.

37.

38.

39.

40.

41.

42.

Exercise Group.

43.

44.

45.

46.

47.

48.

49.

50.

51.

Example 2.2.23. Vertical Stretch/Compression and Reflection about the $x$ -axis.

Example 2.2.25. Horizontal Stretch/Compression and Reflection about the $y$ -axis.