11.2B Graphs of Other Trigonometric Functions
Objectives:
F.TF.5: Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency and midline.
F.EF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F.IF.7: Graph functions expressed symbolically and show key features of the graph by hand in
simple cases and using technology for more complicated cases.
For the Board: You will be able to recognize and graph trigonometric functions.
Anticipatory Set:
1
1
and csc θ =
.
cos θ
sin θ
Since sine and cosine are periodic functions then secant and cosecant are periodic functions.
When cos θ = 0, sec θ will be undefined and have an asymptote.
When sin θ = 0, csc θ will be undefined and have an asymptote.
Recall: sec θ =
Graphing Activity:
Example: Graph y = csc x using a table established from the unit circle.
Example: Graph y = sec x using a table established from the unit circle.
y = csc x
x
0
π/6
π/2
5π/6
π
7π/6
3π/2
7π/4
2π
y = sec x
y
und
2
1
2
und
-2
-1
-2
und
2
1
0
π/2
-1
-1
-2
π
3π/2
2π
x
0
π/3
π/2
2π/3
π
4π/3
3π/2
5π/3
2π
y
1
2
und
-2
-1
-2
und
2
1
2
1
0
π/2
-1
-1
-2
Note: The cosecant function is the visual reciprocal of the sine function.
The secant function is the visual reciprocal of the cosine function.
In y = csc x the asymptotes occur at 0, π, 2π etc. or nπ where n is an integer.
In y = sec x the asymptotes occur at π/2, 3π/2, 5π/2, etc. or π/2 + nπ where n is an integer.
π
3π/2
2π
Characteristics of the Tangent and Cotangent Function
1. Basic ordered pairs:
Cosecant: Reciprocals of the sine function ordered pairs.
Asymptotes: x = 0, π, 2π, or nπ for n an integer
Secant: Reciprocals of the cosine function ordered pairs.
Asymptotes: x = π/2, 3π/2, or π/2 + nπ for n an integer
2. Cosecant: domain: {x|x ≠ nπ where n is an integer}
Secant: domain: {x|x ≠ π/2 + πn where n is an integer}
3. range: {x|x ≥ 1 or x ≤ -1}
4. period: 2
5. no amplitude
The cosecant and secant functions can be transformed.
For the graph of y = a csc bx
a indicates a vertical stretch or compression.
b indicates a horizontal stretch or compression, which changes the period.
2π
The period is
.
b
Guidelines for Sketching the Graphs of Cosecant and Secant Using the 5 Point Method
2π
1. Determine the period of the function. period =
b
2. Set up the table as if you were graphing sine or cosine.
Find the y’s by using the unit circle.
x
0
¼ period
½ period
¾ period
1 period
y
3. Where these functions are 0, secant and cosecant will have asymptotes.
Determine asymptotes.
4. Label the x and y axes with evenly distributed scales.
5. Draw the asymptotes. Plot the points and draw the curve. Think visual reciprocal.
6. Draw additional cycles as needed.
Open the book to page 764 and read example 3.
Example: Using f(x) = cos x as a guide, graph g(x) = ½ sec ( ½ x). Identify1
the period and the asymptotes.
x
y
Period = 2π/( ½ ) = 4 π
0
½
Asymptotes: x = π and x = 3 π
1π
0
0
2π
-½
3π
0
4π
½
-1
π
2π
3π
4π
White Board Activity:
Practice: Using f(x) = sin x as a guide, graph g(x) = 2 csc x. Identify
the period and the asymptotes.
x
y
Period = 2π/1 = 2π
0
0
Asymptotes: x = 0, x = π, and x = 2π
π/2
2
1π
0
3π/2
2
2π
0
Assessment:
Question student pairs.
Assignment:
Text: pgs. 765 prob. 7 – 9, 17 – 19.
4
2
0
π/2
-2
-1
-4
π
3π/2
2π