Understanding Graphing Reciprocal Trigonometric Functions

Trigonometric Function	Reciprocal Trigonometric Function
$y = sin x$	$y = \frac{1}{sin x}$
$y = cos x$	$y = \frac{1}{cos x}$
$y = tan x$	$y = \frac{1}{tan x}$

Trigonometric Function

Reciprocal Trigonometric Function

y = sin x

y = \frac{1}{sin x}

y = cos x

y = \frac{1}{cos x}

y = tan x

y = \frac{1}{tan x}

Properties of $y = a cot b x$
Amplitude	No amplitude
Number of Cycles in $[0, 2 π]$	$2 ∣ b ∣$
Period	$\frac{π}{∣ b ∣}$
Domain	All real numbers except multiples of $\frac{π}{∣ b ∣}$
Range	All real numbers

Properties of

y = a cot b x

Amplitude

No amplitude

Number of Cycles in

[0, 2 π]

2 ∣ b ∣

Period

\frac{π}{∣ b ∣}

Domain

All real numbers except multiples of

\frac{π}{∣ b ∣}

Range

All real numbers

$θ$	$3 cot (\frac{1}{2} θ)$	$y$
$\frac{π}{2}$	$3 cot (\frac{1}{2} \cdot \frac{π}{2})$	$3$
$π$	$3 cot (\frac{1}{2} \cdot π)$	$0$
$\frac{3 π}{2}$	$3 cot (\frac{1}{2} \cdot \frac{3 π}{2})$	$- 3$

θ

3 cot (\frac{1}{2} θ)

y

\frac{π}{2}

3 cot (\frac{1}{2} \cdot \frac{π}{2})

3

π

3 cot (\frac{1}{2} \cdot π)

0

\frac{3 π}{2}

3 cot (\frac{1}{2} \cdot \frac{3 π}{2})

- 3

$x$	$\frac{1}{2} cot 2 x$	$y$
$\frac{π}{8}$	$\frac{1}{2} cot 2 (\frac{π}{8})$	$\frac{1}{2}$
$\frac{π}{4}$	$\frac{1}{2} cot 2 (\frac{π}{4})$	$0$
$\frac{3 π}{8}$	$\frac{1}{2} cot 2 (\frac{3 π}{8})$	$- \frac{1}{2}$

x

\frac{1}{2} cot 2 x

y

\frac{π}{8}

\frac{1}{2} cot 2 (\frac{π}{8})

\frac{1}{2}

\frac{π}{4}

\frac{1}{2} cot 2 (\frac{π}{4})

0

\frac{3 π}{8}

\frac{1}{2} cot 2 (\frac{3 π}{8})

- \frac{1}{2}

Properties of $y = a sec b x$
Amplitude	No amplitude
Number of Cycles in $[0, 2 π]$	$∣ b ∣$
Period	$\frac{2 π}{∣ b ∣}$
Domain	All real numbers except odd multiples of $\frac{π}{2 ∣ b ∣}$
Range	$(- \infty, - ∣ a ∣] \cup [∣ a ∣, \infty)$

Properties of

y = a sec b x

Amplitude

No amplitude

Number of Cycles in

[0, 2 π]

∣ b ∣

Period

\frac{2 π}{∣ b ∣}

Domain

All real numbers except odd multiples of

\frac{π}{2 ∣ b ∣}

Range

(- \infty, - ∣ a ∣] \cup [∣ a ∣, \infty)

$θ$	$2 sec \frac{1}{2} θ$	$y$
$- \frac{π}{2}$	$2 sec (\frac{1}{2} (- \frac{π}{2}))$	$22 \approx 2.83$
$\frac{π}{2}$	$2 sec (\frac{1}{2} \cdot \frac{π}{2})$	$22 \approx 2.83$
$\frac{3 π}{2}$	$2 sec (\frac{1}{2} \cdot \frac{3 π}{2})$	$- 22 \approx - 2.83$
$\frac{5 π}{2}$	$2 sec (\frac{1}{2} \cdot \frac{5 π}{2})$	$- 22 \approx - 2.83$

θ

2 sec \frac{1}{2} θ

y

- \frac{π}{2}

2 sec (\frac{1}{2} (- \frac{π}{2}))

22 \approx 2.83

\frac{π}{2}

2 sec (\frac{1}{2} \cdot \frac{π}{2})

22 \approx 2.83

\frac{3 π}{2}

2 sec (\frac{1}{2} \cdot \frac{3 π}{2})

- 22 \approx - 2.83

\frac{5 π}{2}

2 sec (\frac{1}{2} \cdot \frac{5 π}{2})

- 22 \approx - 2.83

$x$	$2 sec x$	$y$
$- \frac{5 π}{4}$	$2 sec (- \frac{5 π}{4})$	$- 22 \approx - 2.83$
$- \frac{3 π}{4}$	$2 sec (- \frac{3 π}{4})$	$- 22 \approx - 2.83$
$- \frac{π}{4}$	$2 sec (- \frac{π}{4})$	$22 \approx 2.83$
$\frac{π}{4}$	$2 sec \frac{π}{4}$	$22 \approx 2.83$
$\frac{3 π}{4}$	$2 sec \frac{3 π}{4}$	$- 22 \approx - 2.83$
$\frac{5 π}{4}$	$2 sec \frac{5 π}{4}$	$- 22 \approx - 2.83$
$\frac{7 π}{4}$	$2 sec \frac{7 π}{4}$	$22 \approx 2.83$
$\frac{9 π}{4}$	$2 sec \frac{9 π}{4}$	$22 \approx 2.83$

x

2 sec x

y

- \frac{5 π}{4}

2 sec (- \frac{5 π}{4})

- 22 \approx - 2.83

- \frac{3 π}{4}

2 sec (- \frac{3 π}{4})

- 22 \approx - 2.83

- \frac{π}{4}

2 sec (- \frac{π}{4})

22 \approx 2.83

\frac{π}{4}

2 sec \frac{π}{4}

22 \approx 2.83

\frac{3 π}{4}

2 sec \frac{3 π}{4}

- 22 \approx - 2.83

\frac{5 π}{4}

2 sec \frac{5 π}{4}

- 22 \approx - 2.83

\frac{7 π}{4}

2 sec \frac{7 π}{4}

22 \approx 2.83

\frac{9 π}{4}

2 sec \frac{9 π}{4}

22 \approx 2.83

Properties of $y = a csc b x$
Amplitude	No amplitude
Number of Cycles in $[0, 2 π]$	$∣ b ∣$
Period	$\frac{2 π}{∣ b ∣}$
Domain	All real numbers except multiples of $\frac{π}{∣ b ∣}$
Range	$(- \infty, - ∣ a ∣] \cup [∣ a ∣, \infty)$

Properties of

y = a csc b x

Amplitude

No amplitude

Number of Cycles in

[0, 2 π]

∣ b ∣

Period

\frac{2 π}{∣ b ∣}

Domain

All real numbers except multiples of

\frac{π}{∣ b ∣}

Range

(- \infty, - ∣ a ∣] \cup [∣ a ∣, \infty)

$θ$	$2 csc \frac{1}{2} θ$	$y$
$- \frac{3 π}{2}$	$2 csc (\frac{1}{2} (- \frac{3 π}{2}))$	$- 22 \approx - 2.83$
$- \frac{π}{2}$	$2 csc (\frac{1}{2} (- \frac{π}{2}))$	$- 22 \approx - 2.83$
$\frac{π}{2}$	$2 csc (\frac{1}{2} \cdot \frac{π}{2})$	$22 \approx 2.83$
$\frac{3 π}{2}$	$2 csc (\frac{1}{2} \cdot \frac{3 π}{2})$	$22 \approx 2.83$

θ

2 csc \frac{1}{2} θ

y

- \frac{3 π}{2}

2 csc (\frac{1}{2} (- \frac{3 π}{2}))

- 22 \approx - 2.83

- \frac{π}{2}

2 csc (\frac{1}{2} (- \frac{π}{2}))

- 22 \approx - 2.83

\frac{π}{2}

2 csc (\frac{1}{2} \cdot \frac{π}{2})

22 \approx 2.83

\frac{3 π}{2}

2 csc (\frac{1}{2} \cdot \frac{3 π}{2})

22 \approx 2.83

$x$	$\frac{1}{2} csc x$	$y$
$- \frac{5 π}{3}$	$\frac{1}{2} csc (- \frac{5 π}{3})$	$\frac{3}{3} \approx 0.58$
$- \frac{4 π}{3}$	$\frac{1}{2} csc (- \frac{4 π}{3})$	$\frac{3}{3} \approx 0.58$
$- \frac{2 π}{3}$	$\frac{1}{2} csc (- \frac{2 π}{3})$	$- \frac{3}{3} \approx - 0.58$
$- \frac{π}{3}$	$\frac{1}{2} csc (- \frac{π}{3})$	$- \frac{3}{3} \approx - 0.58$
$\frac{π}{3}$	$\frac{1}{2} csc \frac{π}{3}$	$\frac{3}{3} \approx 0.58$
$\frac{2 π}{3}$	$\frac{1}{2} csc \frac{2 π}{3}$	$\frac{3}{3} \approx 0.58$
$\frac{4 π}{3}$	$\frac{1}{2} csc \frac{4 π}{3}$	$- \frac{3}{3} \approx - 0.58$
$\frac{5 π}{3}$	$\frac{1}{2} csc \frac{5 π}{3}$	$- \frac{3}{3} \approx - 0.58$

x

\frac{1}{2} csc x

y

- \frac{5 π}{3}

\frac{1}{2} csc (- \frac{5 π}{3})

\frac{3}{3} \approx 0.58

- \frac{4 π}{3}

\frac{1}{2} csc (- \frac{4 π}{3})

\frac{3}{3} \approx 0.58

- \frac{2 π}{3}

\frac{1}{2} csc (- \frac{2 π}{3})

- \frac{3}{3} \approx - 0.58

- \frac{π}{3}

\frac{1}{2} csc (- \frac{π}{3})

- \frac{3}{3} \approx - 0.58

\frac{π}{3}

\frac{1}{2} csc \frac{π}{3}

\frac{3}{3} \approx 0.58

\frac{2 π}{3}

\frac{1}{2} csc \frac{2 π}{3}

\frac{3}{3} \approx 0.58

\frac{4 π}{3}

\frac{1}{2} csc \frac{4 π}{3}

- \frac{3}{3} \approx - 0.58

\frac{5 π}{3}

\frac{1}{2} csc \frac{5 π}{3}

- \frac{3}{3} \approx - 0.58

$θ$	$5 csc θ$	$y$
$- \frac{3 π}{4}$	$5 csc (- \frac{3 π}{4})$	$- 52 \approx - 7.07$
$- \frac{π}{4}$	$5 csc (- \frac{π}{4})$	$- 52 \approx - 7.07$
$\frac{π}{4}$	$5 csc \frac{π}{4}$	$52 \approx 7.07$
$\frac{3 π}{4}$	$5 csc \frac{3 π}{4}$	$52 \approx 7.07$

θ

5 csc θ

y

- \frac{3 π}{4}

5 csc (- \frac{3 π}{4})

- 52 \approx - 7.07

- \frac{π}{4}

5 csc (- \frac{π}{4})

- 52 \approx - 7.07

\frac{π}{4}

5 csc \frac{π}{4}

52 \approx 7.07

\frac{3 π}{4}

5 csc \frac{3 π}{4}

52 \approx 7.07

Find the period of each reciprocal trigonometric function. Write the answers in exact form.

y = 8 sec (- x)

y = - 9 cot (- 5 x)

The period of a secant function with the form y=asec bx is the quotient of 2π and |b|. Secant Function & Period y = a sec bx & 2π/| b| In our equation we have that b=- 1. Let's substitute this value in the above formula to find the period of y=8sec (- x).

The period of a cotangent function with the form y=acot bx is the quotient of π and |b|. Cotangent Function & Period y=acot bx & π/| b| In our equation we have that b=- 5. Let's substitute this value in the above formula to find the period of y=- 9cot (- 5x).

Which of the following graphs is the graph of $y = \frac{1}{4} cot 2 x ?$

To determine which of the given options corresponds to the graph of y= 14cot 2x, we will draw the graph by ourselves and compare it with the given choices. To graph a cotangent function, the first step is to calculate its period. To do so, consider the general form of a cotangent function. General Form:& y= a cot bx [0.3em] Given Function:& y= 1/4 cot 2x Here, a= 14 and b= 2. The period of the function is the quotient of π and the absolute value of b. Period π/|b| ⇒ π/| 2|=π/2 The period of the function is π2. This means that one cycle goes from 0 to π2, another cycle from π2 to π, and so on. Since the asymptotes occur at the end of each cycle we can draw them on a coordinate plane.

Next, we can divide the period into fourths and locate three equidistant points between the asymptotes. Since a period goes from 0 to π2, we will make a table of values for x= π8, x= π4, and x= 3π8.

x	1/4cot 2x	y=1/4cot 2x
π/8	1/4cot 2( π/8)	1/4
π/4	1/4cot 2( π/4)	0
3π/8	1/4cot 2( 3π/8)	- 1/4

The points we found in the table are ( π8, 14), ( π4,0), and ( 3π8,- 14). Let's plot these three points on our the plane.

We can connect these points to graph one period of the given function.

Finally, let's draw all the cycles for values of x between - π and π.

This graph matches choice B.

Which of the following graphs is the graph of $y = \frac{1}{2} sec x ?$

To determine which of the given options corresponds to the graph of y= 12sec x, we will draw the graph by ourselves and compare it with the given choices. To graph a secant function, the first step is to calculate the period. Let's consider the general form of a secant function and the given function. General Form y= asec bx [0.5em] Given Function y=1/2sec x ⇔ y= 1/2sec 1x Here, we see that a= 12 and b= 1. We know that the period of the function is the quotient of 2π and the absolute value of b. Period 2π/|b| ⇒ 2π/| 1|=2π Now that we know that the period of the function is 2π, let's consider the related cosine function. cc Secant & Related Cosine Function & Function [0.5em] y=1/2sec x & y=1/2cos x We can now graph this cosine function on a coordinate plane for values of x between - π2 and 3π2.

The asymptotes of the secant function occur at the zeros of the cosine function. Furthermore, the maximum and minimum points of the cosine function are also points on the curve of the secant function.

Next, we can find more points on the curve of y=2sec x by making a table of values.

x	1/2sec x	y
- π/4	1/2sec ( - π/4)	sqrt(2)/2≈ 0.7
π/4	1/2sec π/4	sqrt(2)/2≈ 0.7
3π/4	1/2sec 3π/4	- sqrt(2)/2≈ - 0.7
5π/4	1/2sec 5π/4	- sqrt(2)/2≈ - 0.7

We can plot the points we found in the table. Finally, we can connect each set of points with smooth curves.

This graph corresponds to choice A.

Which of the following graphs is the graph of $y = - \frac{1}{4} csc x ?$

To determine which of the given options corresponds to the graph of y=- 14csc x, we will draw the graph by ourselves and compare it with the given choices. To graph a cosecant function, the first step is to calculate the period. Let's consider the general form of a cosecant function and the given function. General Form y= acsc bx [0.5em] Given Function y=- 1/4csc x ⇔ y=- 1/4csc 1x Here, a= - 14 and b= 1. The period of the function is the quotient of 2π and the absolute value of b. Period 2π/|b| ⇒ 2π/| 1|=2π The period of the function is 2π. Let's now consider the related sine function. cc Cosecant & Related Sine Function & Function [0.5em] y=- 1/4csc x & y=- 1/4sin x We can graph this sine function on a coordinate plane for values of x between - π and π.

The asymptotes of the cosecant function occur at the zeros of this sine function. Furthermore, the maximum and minimum points of the sine function are also points on the curve of the cosecant function. Let's show this in our diagram.

Next, we can find more points on the curve of y=- 14csc x by making a table of values.

x	- 1/4csc x	y
- 3π/4	- 1/4csc ( - 3π/4)	sqrt(2)/4≈ 0.35
- π/4	- 1/4csc ( - π/4)	sqrt(2)/4≈ 0.35
π/4	- 1/4csc π/4	- sqrt(2)/4≈ - 0.35
3π/4	- 1/4csc 3π/4	- sqrt(2)/4≈ - 0.35

We can now plot the points we found in the table. Finally, we can connect the sets of points with smooth curves.

This graph matches choice D.

Ramsha wants to find the least positive solution to the following equation.

2 csc x = 2

Solve the equation by graphing and find the value of

x

that Ramsha is looking for. Write the answer in radians rounded to three significant figures.

To solve the equation graphically, we will graph the cosecant function whose function rule is represented by the left-hand side of the equation. The first step is to calculate the period of the function. Consider the general form of a cosecant function and our function. General Form y= acsc bx [0.5em] Our Function y=2csc x ⇔ y= 2csc 1x Here, a= 2 and b= 1. The period of the function is the quotient of 2π and | b|. Period 2π/|b| ⇒ 2π/| 1|=2π The period of the function is 2π. Let's now consider the related sine function. cc Cosecant & Related Sine Function & Function [0.5em] y=2csc x & y=2sin x We can graph this sine function on a coordinate plane. Since we are looking for a positive value of x, we will not consider the negative values of the x-axis.

The asymptotes of the cosecant function occur at the zeros of the sine function. Furthermore, the maximum and minimum points of the sine function are also points on the curve of the cosecant function.

Next, we can find more points on the curve of y=2csc θ by making a table of values.

θ	2 csc θ	y
π/4	2csc π/4	2sqrt(2)≈ 2.8
3π/4	2csc 3π/4	2sqrt(2)≈ 2.8
5π/4	2csc 5π/4	- 2sqrt(2)≈ - 2.8
7π/4	2csc 7π/4	- 2sqrt(2)≈ - 2.8

Finally, let's plot the points we found in the table. We will also connect each set of points with smooth curves.

Now that we have graphed the function, we can locate the point whose x-coordinate is the least positive number and whose y-coordinate is 2.

We can see that the value of x that Ramsha is looking for is π2. This number, rounded to three significant figures, is 1.57. π/2 = 1.570796 ... ⇔ π/2 ≈ 1.57

Graphing Reciprocal Trigonometric Functions

Catch-Up and Review

Communication Tower

Exploring the Graphs of the Cosine and Sine Functions, and Their Quotient

The Cotangent Function

Graphing a Cotangent Function

Extra

Tanabata Trees and a Cotangent Function

Answer

Hint

Solution

Exploring the Graphs of the Cosine Function and its Reciprocal Function

The Secant Function

Graphing a Secant Function

A Bowl and a Secant Function

Answer

Hint

Solution

Exploring the Graphs of the Sine Function and its Reciprocal Function

The Cosecant Function

Graphing a Cosecant Function

An Umbrella and a Cosecant Function

Answer

Hint

Solution

Finding the Period

Drawing and Using the Graph of a Cosecant Function

Hint

Solution

Graphing Reciprocal Trigonometric Functions

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