Name: ______________________________
Algebra II Guided Notes: Chapter 14 – Trigonometric Graphs and Equations
Section 1 – Graph Sine, Cosine, and Tangent Functions
Graphing the Cosine Function:
2π
π
π
π
π
θ
0
6
4
3
2
3
cos θ
1
y = cos θ
3π
5π
4
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
3
2
Graphing the Sine Function:
π
π
π
π
θ
0
6
4
3
2
2π
3
y = sin θ
3π
5π
4
6
π
7π
6
Domain:
Range:
Maximum:
Minimum:
y-intercept
x-intercept(s)
Amplitude:
Period:
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
sin θ
Domain:
Range:
Maximum:
Minimum:
y-intercept
x-intercept(s)
Amplitude:
Period:
1
Graphing the Tangent Function:
θ
0
π
π
π
π
6
4
3
2
2π
3
y = tan θ
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
tan θ
Domain:
Range:
Maximum:
Minimum:
y-intercept
x-intercept(s)
Amplitude:
Period:
Amplitude: ______________________________________________________________________________
Periodic:
______________________________________________________________________________
Cycle:
______________________________________________________________________________
Period:
______________________________________________________________________________
Sketch the following functions on the axes below. Set the window of your graphing
π
calculator to: xmin = − 2π , xmax = 2π , xscl = , ymin = -6, ymax = 6, yscl = 1. You need to
2
be in radians. After graphing each equation, fill in the table below.
y = sin x
y = 5 sin x
y=
1
sin x
2
Period
Minimum
Maximum
Range
y-intercept
x-intercept(s)
2
Sketch the following functions on the axes below. Set the window of your graphing
calculator to: xmin = − 6π , xmax = 6π , xscl = π , ymin = -2, ymax = 2, yscl = 1.
After graphing each equation, fill in the table below.
y = cos x
y = cos 2 x
⎛ x⎞
y = cos⎜ ⎟
⎝3⎠
Period
Minimum
Maximum
Range
y-intercept
x-intercept(s)
Amplitude and Period
The amplitude and period of the graphs of y = a sin bx and y = a cos bx , where a and b are
nonzero real numbers, are as follows:
Amplitude: _________
Period: __________
The period and vertical asymptotes of the graph of y = a tan bx , where a and b are nonzero
real numbers, are as follows:
Period: ________
Vertical Asymptotes: ___________________________
3
Determine the amplitude and period of each function.
Amplitude
Period
1
a.) y = sin (2 x )
4
⎛ x⎞
b.) y = 3 cos⎜ ⎟
⎝2⎠
⎛ πx ⎞
c.) y = 5 sin ⎜ ⎟
⎝ 4⎠
d.) y =
1
cos(3 x )
2
Graph each function.
a.
y = 3 cos x
b.
y = − sin 3θ
4
c.
y = 5 cos 4θ
d.
y = 2 tan x
Frequency: ______________________________________________________________________________
Example
A sound consisting of a single frequency is called a pure tone. An audiometer produces
pure tones to test a person’s auditory functions. Suppose an audiometer produces a pure
tone with a frequency f of 2000 hertz (cycles per second). The maximum pressure P
produced from the pure tone is 2 millipascals. Write and graph a sine model that gives the
pressure P as a function of the time t (in seconds).
5
Section 2 – Translate and Reflect Trigonometric Graphs
Given the general form, y = a sin b( x − h ) + k or y = a cos b( x − h ) + k :
Amplitude
___________
Horizontal Shift ___________
Period
__________
Vertical Shift
__________
Examples: Graph each equation.
1. y = cos(θ + π )
Amplitude
Period
Phase Shift
Vertical Shift
π⎞
⎛
2. y = sin 4⎜θ − ⎟
4⎠
⎝
Amplitude
Period
Phase Shift
Vertical Shift
3. y = −3 sin θ + 2
Amplitude
Period
Phase Shift
Vertical Shift
6
4. y = 2 cos
1
(x + π ) − 1
4
Amplitude
Period
Phase Shift
Vertical Shift
1⎛
π⎞
5. y = 3 sin ⎜ x − ⎟ + 1
2⎝
2⎠
Amplitude
Period
Phase Shift
Vertical Shift
π⎞
⎛
6. y = 2 cos⎜θ + ⎟ − 3
2⎠
⎝
Amplitude
Period
Phase Shift
Vertical Shift
7
Example
An outboard motor rotates its propeller at 5000 revolutions per minute. The center of the
propeller is 18 inches below the surface of the water. The depth D(t) (in feet) of a point on
the propeller at time t (in seconds) can be modeled by the equation
⎛ 500π ⎞
D (t ) = 5 cos⎜
t ⎟ + 18 . Graph the depth of the point below water as a function of time.
⎝ 3 ⎠
Where are the maximum and minimum depths?
Section 4 – Solve Trigonometric Equations
Example 1
Solve each equation for all real values.
a.
2 cos x + 1 = 0
b.
2 sin x + 4 = 5
Example 2
Solve each equation in the interval 0 ≤ x < 2π .
a.
3 − tan 2 x = 0
b.
3 csc 2 x = 4
8
c.
2sin2x + sinx – 1 = 0
d.
1
sin x cos x − cos x = 0
2
Example 3
What is the general solution of 2cos3x – cosx = 0?
Example 4
Solve sin2x – 4sinx + 1 = 0 in the interval 0 ≤ x < π .
9
Example 5
Suppose the average monthly high temperature (oF) T in a Midwestern U.S. city can be
2π ⎞
⎛π
modeled by T = 27 sin ⎜ t −
⎟ + 57 when t is in months and t = 1 corresponds to January. In
3 ⎠
⎝6
which month is the average high temperature 84oF?
Section 5 – Write Trigonometric Functions and Models
Graphs of sine and cosine functions are called sinusoids.
To write equations of sinusoidal functions, use y = a sin b( x − h ) + k or y = a cos b(x − h ) + k .
Recall that the amplitude is _________ and the period is __________.
Example 1
Write a function for the sinusoids below.
a.)
b.)
c.)
10
Example 2
A physical therapist recommends an exercise for a patient recovering from shoulder surgery.
The patient extends his arm and rotates it in a circular motion. At the highest point in the
circle, the patient’s arm is 65 inches above the floor. His arm is 40 inches above the floor at
the circle’s lowest point. His arm makes 4 revolutions per second. Write a model for the
height (in inches) of the arm as a function of the time t (in seconds) if the arm is at its highest
point when t = 0.
Example 3
The table below shows the number of sunlight hours h for each month during one year in
Anchorage, Alaska. The time t is measured in months with t = 1 representing January. Write
a trigonometric model that gives h as a function of t.
t
h
1
5.7
2
7.8
3
10.4
4
13.4
5
16.9
6
18.7
7
19.2
8
17.1
9
14.3
10
11.4
11
8.5
12
6.1
Example 4
Use a graphing calculator to write a sine model that gives the average daily temperature T
(in degrees Fahrenheit) for Boston, Massachusetts, as a function of the time t (in months),
where t = 1 represents January.
t
T
1
29
2
32
3
39
4
48
5
59
6
68
7
74
8
72
9
65
10
54
11
45
12
35
11