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

To find the length of the wire that makes an angle of

\frac{π}{4}

radians with the ground, graph the cosecant function. The first step is to calculate the period of the function. Consider the general form of a cosecant function and the given function.

General Form y = a csc b θ Given Function y = 5 csc θ \Leftrightarrow y = 5 csc 1 θ

Here,

a = 5

and

b = 1 .

The period of the function is the quotient of

2 π

and

∣ b ∣ .

Period \frac{2 π}{∣ b ∣} \Rightarrow \frac{2 π}{∣ 1 ∣} = 2 π

The period of the function is

2 π .

Now consider the related sine function.

Cosecant Function y = 5 csc θ Related Sine Function y = 5 sin θ

This sine function can be graphed on a coordinate plane.

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, more points on the curve of

y = 5 csc θ

will be found by making a table of values.

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

Finally, the points found in the table are be plotted and each set of points connected with smooth curves.

Now that the function has been graphed, the point at $θ = \frac{π}{4}$ can be located and its $y -$ coordinate identified.

The exact $y -$ coordinate cannot be found in the graph, but its nearest integer can be identified as $7 .$ Therefore, the length of the wire that makes an angle of $\frac{π}{4}$ radians with the ground is about $7$ meters.

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Catch-Up and Review

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The Cotangent Function

Graphing a Cotangent Function

Extra

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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

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