Unit 6 – Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5)

Unit 6 – Introduction to Trigonometry
Graphing Other Trig Functions
(Unit 6.5)
William (Bill) Finch
Mathematics Department
Denton High School
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Lesson Goals
When you have completed this lesson you will:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
2 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Lesson Goals
When you have completed this lesson you will:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
2 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Lesson Goals
When you have completed this lesson you will:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
2 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Lesson Goals
When you have completed this lesson you will:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
2 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Overview
The tangent and cotangent functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
3 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Overview
The secant and cosecant functions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
4 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
The Parent Tangent Function
Domain: x ∈ <, x 6= π2 + nπ
Range: (−∞, ∞)
x-intercept: nπ
y-intercept: 0
Continuity: inf. discont. at
x = π2 + nπ
Asymptotes: x = π2 + nπ
Symmetry: origin (odd
function)
Extrema: none
End behavior: does not exist
W. Finch
Graph Other Trig Functions
DHS Math Dept
5 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Period of the Tangent Function
One period of the tangent
function is π.
For y = a tan(bx + c) + d the
period is
period =
W. Finch
Graph Other Trig Functions
π
|b|
DHS Math Dept
6 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 1 – Horizontal Dilation of Tangent
Locate the vertical asymptotes and then sketch the graph of
π
y = tan x.
3
W. Finch
Graph Other Trig Functions
DHS Math Dept
7 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 2 – Reflections and Translations of
Tangent
Locate the vertical asymptotes and sketch the graph of
π
y = − tan x
4
W. Finch
Graph Other Trig Functions
DHS Math Dept
8 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 3 – Reflections and Translations of
Tangent
Locate theverticalasymptotes and sketch the graph of
π
y = − tan x +
2
W. Finch
Graph Other Trig Functions
DHS Math Dept
9 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
The Parent Cotangent Function
Domain: x ∈ <, x 6= nπ
Range: (−∞, ∞)
π
x-intercept: + nπ
2
y-intercept: none
Continuity: inf. discont. at
x = π2 + nπ
Asymptotes: x = nπ
Symmetry: origin (odd
function)
Extrema: none
End behavior: does not exist
W. Finch
Graph Other Trig Functions
DHS Math Dept
10 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Period of the Cotangent Function
One period of the cotangent
function is π.
For y = a cot(bx + c) + d the
period is
period =
W. Finch
Graph Other Trig Functions
π
|b|
DHS Math Dept
11 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 4
Locate the vertical asymptotes and sketch the graph of
y = cot 2x.
W. Finch
Graph Other Trig Functions
DHS Math Dept
12 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
The Parent Secant Function
Domain: x ∈ <, x 6= π2 + nπ
Range: (−∞, −1] ∪ [1, ∞)
x-intercept: none
y-intercept: 1
Continuity: inf. discont. at
x = π2 + nπ
Asymptotes: x = 2ı + nπ
Symmetry: y -axis (even
function)
End behavior: does not exist
W. Finch
Graph Other Trig Functions
DHS Math Dept
13 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
The Parent Cosecant Function
Domain: x ∈ <, x 6= nπ
Range: (−∞, −1] ∪ [1, ∞)
x-intercept: none
y-intercept: 1
Continuity: inf. discont. at
x = nπ
Asymptotes: x = nπ
Symmetry: origin (odd
function)
End behavior: does not exist
W. Finch
Graph Other Trig Functions
DHS Math Dept
14 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 5
Locate the vertical asymptotes and sketch the graph of
y = − sec 2x
W. Finch
Graph Other Trig Functions
DHS Math Dept
15 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 6
Locate the
asymptotes and sketch the graph of
vertical
π
y = csc x +
3
W. Finch
Graph Other Trig Functions
DHS Math Dept
16 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Damped Trigonometric Functions
Damped oscillation results when a sinusoid is multiplied by a
function f (x) so the amplitude of the sinusoid is reduced as x
approaches ±∞ or as x approaches 0 from both directions.
W. Finch
Graph Other Trig Functions
DHS Math Dept
17 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 7
Identify the damping factor f (x). Then sketch a graph of the
function, f (x) and −f (x). Include the viewing window from
the calculator.
x
y = sin x
2
W. Finch
Graph Other Trig Functions
DHS Math Dept
18 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 8
Identify the damping factor f (x). Then sketch a graph of the
function, f (x) and −f (x). Include the viewing window from
the calculator.
y = x 2 cos 3x
W. Finch
Graph Other Trig Functions
DHS Math Dept
19 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Damped Harmonic Motion
When the amplitude of the motion of an object decreases with
time due to friction, the motion is called damped harmonic
motion.
y = ke −ct sin ωt
y = ke −ct cos ωt
where c > 0 and is the
damping constant, k is the
displacement, t is time, and ω
is the period.
W. Finch
Graph Other Trig Functions
DHS Math Dept
20 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
Example 9
A guitar string is plucked at a distance of 0.95 centimeters
above its rest position, then released, causing a vibration. The
damping constant for the string is 1.3, and the note produced
has a frequency of 200 cycles per second.
a) Write a trig function that models the motion of the string.
b) Determine the amount of time t that it takes the string to
be damped so that −0.38 ≤ y ≤ 0.38.
W. Finch
Graph Other Trig Functions
DHS Math Dept
21 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
What You Learned
You can now:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
I
Do problems Chap 4.5 #1-15 odd, 17, 23, 27, 29-31 odd
W. Finch
Graph Other Trig Functions
DHS Math Dept
22 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
What You Learned
You can now:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
I
Do problems Chap 4.5 #1-15 odd, 17, 23, 27, 29-31 odd
W. Finch
Graph Other Trig Functions
DHS Math Dept
22 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
What You Learned
You can now:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
I
Do problems Chap 4.5 #1-15 odd, 17, 23, 27, 29-31 odd
W. Finch
Graph Other Trig Functions
DHS Math Dept
22 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
What You Learned
You can now:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
I
Do problems Chap 4.5 #1-15 odd, 17, 23, 27, 29-31 odd
W. Finch
Graph Other Trig Functions
DHS Math Dept
22 / 22
Introduction
Tangent
Cotangent
Secant / Cosecant
Damped
Summary
What You Learned
You can now:
I
Graph the parent tangent, cotangent, secant, and
cosecant funtions.
I
Graph transformations of these functions.
I
Graph damped trigonometric functions.
I
Do problems Chap 4.5 #1-15 odd, 17, 23, 27, 29-31 odd
W. Finch
Graph Other Trig Functions
DHS Math Dept
22 / 22