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

Consider the following cotangent function.

y = 5 cot b x

It is known that the period of this function is

\frac{1}{2} .

Find all the possible values for

b .

Write these values in their exact form.

Let's start by recalling that the period of a cotangent function is the quotient of π and the absolute value of b. Period: π/|b| We know that the period of the given function is 12. Therefore, the above quotient is equal to 12.

We found that the absolute value of b is 2π. Therefore, b can be either - 2π or 2π. |b|=2π ↙ ↘ b=- 2π or b=2π

Order the functions from the least average rate of change to the greatest average rate of change over the interval $- \frac{π}{4} \leq x \leq \frac{π}{4} .$

We are asked to order the given functions with respect to their average rate of change over the interval - π4 ≤ x ≤ π4. First, let's recall the general formula for the average rate of change. Average Rate of Change = f(x_2)-f(x_1)/x_2-x_1 Here, x_1 and x_2 are the endpoints of the interval, and f(x_1) and f(x_2) are the values of the function at these points. We are going to calculate the average rate of change for each function one at a time.

Function A

We want to evaluate the average rate of change over the interval - π4 ≤ x ≤ π4. To do that, let's first add the endpoints of the interval to the graph.

To calculate the average rate of change in the desired interval, we will use the points (- π4, - 2) and ( π4,2). We can substitute the coordinates of these point into the formula for the average rate of change and then evaluate the resulting numeric expression.

The average rate of change of the function in choice A is 8π.

Functions B, C, and D

By following the same steps, we can evaluate the average rate of change of the remaining functions.

	Endpoints	Substitute	Evaluate
Function A	( - π/4, - 2) and ( π/4, 2)	2-( - 2)/π4-( - π4)	8/π≈ 2.55
Function B	( - π/4, 1) and ( π/4, - 1)	- 1- 1/π4-( - π4)	- 4/π≈ - 1.27
Function C	( - π/4, - 1/2) and ( π/4, 1/2)	12-( - 12)/π4-( - π4)	2/π≈ 0.64
Function D	( - π/4, 2) and ( π/4, - 2)	- 2- 2/π4-( - π4)	- 8/π≈ -2.55

Conclusion

We can now order the functions from the least average rate of change to the greatest average rate of change.

	Average Rate of Change
Value	- 8/π	- 4/π	3/π	8/π
Function	D	B	C	A

Ramsha made a mistake when finding the period of $y = cot 5 x .$

Correct the error and find the period of the function. Write the answer in exact form.

The period of a cotangent function with the form y=a cot bx is the quotient of π and |b|. Cotangent Function & Period y = a cot bx & π/| b| In our function, we know that b=5. Let's substitute this number in the above formula to find the period of y=cot 5x.

The period of the function is π5. The mistake Ramsha made was using the incorrect formula. She thought that the period of a cotangent function was 2π|b| instead of π|b|.

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

Function A

Functions B, C, and D

Conclusion

Graphing Reciprocal Trigonometric Functions

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