PRECALCULUS
CHAPTER 4B
TRIGONOMETRY
PRANGE/JOHNSON
1
Blank page
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Trigonometry Chapter 4B
Learning Target
Practice for the
Learning Target
4.5.1
I can graph sine functions.
worksheet in
this packet
4.5.2
I can graph cosine functions.
worksheet in
this packet
4.5.3
I can graph sine and cosine functions with a
vertical translation.
worksheet in
this packet
4.6.1
I can graph tangent functions including
vertical translations.
worksheet in
this packet
4.8.1
I can use periodic functions to model and
solve real world problems.
worksheet in
this packet
Score on
Learning
Target Quiz
Help
needed?
yes/no
Essential Questions for the chapter:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How are the six trigonometric and circular functions related to each other?
Essential Questions for the course
1.
2.
3.
4.
5.
6.
7.
How is this similar or different from what I have done before?
What can I do to retain what I have learned?
Does my answer make sense? If not, what do I do?
Do I need help, and where do I go to find it?
How would a calculator make this problem easier to do?
How do I explain or justify my work to myself and others?
What is the given information and how do I use it?
3
LEARNING TARGET QUIZ SCORING RUBRIC
4 MASTERY
I completely understand the strategy and mathematical operations to be used,
and I used them correctly.





My work shows what I did and what I was thinking while I worked the problem.
The way I worked the problem makes sense and is easy for someone else to follow.
I followed through with my strategy from beginning to end.
My explanation and work was clear and organized.
I did all of my calculations correctly.
3
DEVELOPING MASTERY
I completely understand the strategy and mathematical operations to be used,
but a minor error kept me from completing the problem correctly.
2
BASIC UNDERSTANDING
I used mathematical operations and a strategy that I think works for most of the
problem.





1
INCOMPLETE UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.

0
Someone might have to add information for my explanation to be easy to follow.
I know which operations I should have used, but couldn’t complete the problem.
I think I know what the problem is about, but I might have a hard time explaining it.
I’m not sure how much detail I need in order to help someone understand what I did.
I made several calculation errors.
I tried several things, but didn’t get anywhere.
NO EVIDENCE
I left the problem blank.



I didn’t know how to begin.
I don’t know what to write.
I provided no evidence of understanding.
4
4-5 Investigation – You need your textbook. The purpose of this investigation is to help you remember
the graphic effect of each of the constants a, b, c, d in equations of the form:
y  d  a sin( bx  c)
Set your graphing calculator to this window: Xmin = -5
Xmax = 5
Xscl = 1
Ymin = -5
Ymax = 5
Yscl = 1
Use your graphing calculator to graph y  a sin x for each of the following a values. Sketch the graph
of each function on the coordinate plane using the identified color for each graph.
1
2
1. Red a  1 (parent function)
2. Blue a 
3. Orange a  2
4. Purple a  3
How does the value of a affect the shape of the graph? ____________________________
______________________________________________________________________
5
Use your graphing calculator to graph y  a sin x for each of the following a values. Sketch the graph
of each function on the coordinate plane using the identified color for each graph.
1
2
1. Red a  1 (parent function)
2. Blue a  
3. Orange a  2
4. Purple a  3
What happens to the graph if a is negative?________________________________________
____________________________________________________________________________
Summarize the idea you discovered in these exercises.
If a > 0the basic sine shape looks like
If a < 0 the basic sine shape looks like
The absolute value of a is called the ________________________ of the function.
6
Use your graphing calculator to graph y  sin( x  c) for each of the following c values. Sketch the
graph of each function on the coordinate plane using the identified color for each graph.
1. Red c  0 (parent function)
2. Blue c 
1
2
3. Orange a  2
How does the value of c affect the shape of the graph? ____________________________
______________________________________________________________________
7
Use your graphing calculator to graph y  sin( x  c) for each of the following c values. Sketch the
graph of each function on the coordinate plane using the identified color for each graph.
1. Red c  0 (parent function)
2. Blue c  
1
2
3. Orange c  2
Summarize the idea you discovered in these exercises.
If the equation reads y  sin( x  c) , the graph shifts __________________ to the
________________.
If the equation reads y  sin( x  c) the graph shifts ___________________ to the
________________.
8
Use your graphing calculator to graph y  d  sin( x) for each of the following d values. Sketch the
graph of each function on the coordinate plane using the identified color for each graph.
1. Red d  0 (parent function)
2. Blue d  3
3. Orange d  3
How does the value of d affect the shape of the graph? ____________________________
______________________________________________________________________
Summarize the idea you discovered in these exercises.
If d is positive, the graph shifts ___________________ in the ________________ direction.
If d is negative the graph shifts ___________________ in the ________________ direction.
9
Use your graphing calculator to graph y  sin bx for each of the following b values. Sketch the graph
of each function on the coordinate plane using the identified color for each graph.
1. Red b  1 (parent function)
2. Blue b 
1
2
3. Orange b  2
How does the value of b affect the graph? _______________________________________
__________________________________________________________________________
Summarize the idea you just discovered in this activity.
One complete cycle of the graph, before it repeats again, is called a __________________.
Horizontal stretching affects the _________________of your graph.
If 0  b  1, the graph horizontally ________________.
If b  1 , the graph horizontally ________________.
10
4-5 Warm Up(s)
2
1. Sketch f ( x)  sin x
3
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
11
Date _________
Notes: 4-5
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can graph sine functions.
Vocabulary:
amplitude
period
beginning point
interval length
The graphs of trigonometric functions have a repeating pattern which means that they are _________________. The
portion that repeats in one period is called a ______________, and the height of the graph is known as the
______________.
a) First fill in the coordinates for 0,
(
(
(
,
(
(
,
6
,
,
3
,
2 5
3
,
, ,
, 2 on the circle below.
2 3
6
2
,
)
)
(
)
,
,
  
)
,
)
(
,
(
)
,
(
(
,
)
)
,
)
)
b) Now, plot the sine value (y-coordinate) for each x value below and draw the graph.
1





6
6
3
2
2 5
3 6

7
6
4
3
3
2
5 11
2
3
6
-1
The standard form of the sine function is _____________________________. The amplitude is
________, the length of the period is ___________, with the beginning point of the interval at
____________. To find the four intervals, divide the period by ________ then add to the beginning
point. Let’s find the amplitude, period length, interval length for the graph above.
12
Example 1:
Sketch f ( x)  3sin x
3




4
4
2
3
4

5
4
3 7
2 4
2
-3
Amp =
Example 2:
Period =
I.L. =
B.P. =
Key points at:
Sketch f ( x)  sin 2 x
Amp =
Period =
I.L. =
B.P. =
Key points at:
13
Example 3:
Sketch f ( x)  sin  x 



4
Amp =
Example 4:
Period =
I.L. =
B.P. =
Key points at:
Sketch f ( x)  2sin(3 x   )
Amp =
Period =
I.L.=
B.P.=
Key points at:
14
Example 5:
Sketch f ( x)  3sin  2 x 



4
Amp =
Example 6:
Period =
I.L. =
B.P. =
Key points at:
1 

Sketch f ( x)   sin  x  
3 
2
**Not Typical:
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
15
Example 7:
Sketch f ( x)  2sin( x   )
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
16
4-5 Sine Homework
1. Sketch f ( x)  2.5sin x
Amp =
Period =
I.L. =
B.P. =
Key points at:
2. Sketch f ( x)  sin 3x
Amp =
Period =
I.L. =
B.P. =
Key points at:
17
3. Sketch f ( x)  sin  x 



2
Amp =
Period =
I.L. =
B.P. =
Key points at:
4. Sketch f ( x)  sin(4 x   )
Amp =
Period =
I.L.=
B.P.=
Key points at:
18
5. Sketch f ( x)  2sin  5x   
Amp =
Typical?
Period =
I.L. =
B.P. =
Key points at:
6. Sketch f ( x)  5sin  4 x 



2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
19
4 – 5 Warm Up(s)
20
Date _______
Notes: 4-5
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can graph cosine functions.
Vocabulary:
amplitude
period
beginning point
interval length
The graphs of trigonometric functions have a repeating pattern which means that they are _________________. The
portion that repeats in one period is called a ______________, and the height of the graph is known as the
______________.
a) First fill in the coordinates for 0,
(
(
(
,
(
(
,
6
,
,
3
,
2 5
3
,
, ,
, 2 on the circle below.
2 3
6
2
,
)
)
(
)
,
,
  
)
,
)
(
,
(
)
,
(
(
,
)
)
,
)
)
b) Now, plot the sine value (y-coordinate) for each x value below and draw the graph.
1





6
6
3
2
2 5
3 6

7
6
4
3
3
2
5 11
2
3
6
-1
The standard form of the sine function is _____________________________. The amplitude is
________, the length of the period is ___________, with the beginning point of the interval at
____________. To find the four intervals, divide the period by ________ then add to the beginning
point. Let’s find the amplitude, period length, interval length for the graph above.
21
Example 1:
Sketch f ( x)  2cos 4 x

Example 2:



8
8
4
3
8
5
8

2
3 7
4 8

Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
Sketch f ( x)  3cos  2 x 



4
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
22
Example 3:
Example 4:
Sketch f ( x)  4 cos  x 
1
2


4
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
Sketch f ( x)  cos  x 
1
2



4
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
23
Example 5:
Example 6:
Sketch f ( x)  7cos  4 x   
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at
Sketch f ( x)  5cos  3 x 



2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at
24
Example 7:
Sketch f ( x)   cos(2 x   )
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
25
4-5 Cosine Homework
1. Sketch f ( x)  3cos 2 x
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
2. Sketch f ( x)  2 cos  4 x 



2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
26
3. Sketch f ( x)  3cos  x 
1
4


2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
4. Sketch f ( x)  cos  x 
1
4



2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
27
5. Sketch f ( x) 
3


cos  2 x  
4
2

Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at
6. Sketch f ( x)   cos  4 x 



2
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at
28

1
7. Sketch f ( x)  3cos  x  
2
3
Amp =
Typical?
Period =
I.L.=
B.P.=
Key points at:
29
4-5 Warm Up(s)
30
Date _______
Notes: 4-5
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can graph sine and cosine function with a vertical translation.
Vocabulary:
Amplitude
period
beginning point
interval length
vertical translation
Trigonometric Vertical Translations: When a constant is added to a trigonometric function, the
graph is shifted upward or downward. If (x,y) are coordinates of y  sin x , the ( x, y  d ) are
coordinates of y  sin x  d .
A new horizontal axis called the _________________ becomes the reference line about which the
graph oscillates. For the graph of y  sin   d the midline is the graph of __________________.
Example 1: Graph y  sin x  2
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
31
Example 2: Graph y  2  cos  
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
32
Example 3: Graph y  2sin  2 x     4
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
33
 
 
Example 4: y  3sin  2  x 
 
 1
2  
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
34
1 
2
Example 5: Graph y  6  4cos    
 

3  
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
35
4-5 Sine and Cosine with Vertical Translation Homework
1. Graph y  sin x  3
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
36
2. Graph y  4  cos  
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
37
3. Graph y  2sin  x     1
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
38


4. Graph y  1  3cos    
4

Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
39
5. y  3cos  2 x     2
Amp =
Typical?
Period =
I.L.=
B.P.=
V. T. =
Key points at
40
4-6 Warm Up(s)
41
Date _______
Notes: 4-6
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can graph tangent functions.
Vocabulary:
Amplitude
Midpoint
period
vertical asymptote
tan:

Angle measure:
Value

beginning point
vertical translation

2
______

0
4
______
______
interval length


4
2
______
______
Now, plot the tangent value (y-coordinate) for each x value below and draw the graph.
1


2




4
4
2
3
4

5
4
3
2
7
4
2
-1
MEMORIZE
The standard form of the tangent function is _____________________________. There is no
amplitude but the graph does have a curve at ________, the length of the period is ___________,
with the midpoint of the period at ____________. To find the four intervals, divide the period by
________. To find the asymptotes add the interval twice to the midpoint and subtract the interval
twice from the midpoint.
42
Example 1:


Sketch f ( x)  tan  x  
4

Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
43
Example 2:
Sketch f ( x)  2 tan( x   )
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
44
Example 3: Sketch f ( x)  3 tan  x 



2
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
45
Example 4:
Sketch f ( x)  2 tan( )
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
46
Example 5:


Sketch f ( x)   tan    
8

Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
47
Example 6:


Sketch f ( x)  2 tan      1
4

Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
48
4-6 Tangent Homework


1. Sketch f ( x)  tan  x  
4

Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
49
2. Sketch f ( x) 
1
tan( x   )
2
Curve
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
50
3. Sketch f ( x)  2 tan  x 



2
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
51


4. Sketch f ( x)   tan    
2

Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
52

1
5. Sketch f ( x)   tan    
4
4
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
53

1
6. Sketch f ( x)  tan      2
4
8
Curve at
Typical?
Period =
I.L. =
M.P. =
V.T. =
Stretch =
Key points at
54
4-8 Warm Up(s)
55
Date _________
Notes: 4-8
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use periodic functions to model and solve real world problems.
Vocabulary:
amplitude
period
beginning point
interval length
vertical translation
1. Given that a wheel is traveling along a level path and a green dot is on that wheel. The diameter of
the wheel is two feet. The wheel completes one revolution every five seconds. Let the horizontal axis
represent time and the vertical axis represent how far off the ground the green dot is, sketch a graph of
what this may look like. (NEED AN APPLET FOR THIS!)
56
2. Suppose you are riding a Ferris wheel and you walk up the stairs to sit in the bottom car. The bottom
car is 8 feet off the ground. The diameter of the Ferris wheel is 50 feet. It takes 30 seconds to complete
one revolution. Let the horizontal axis represent time and the vertical axis represent how far off the
ground you are, sketch a graph of what this may look like. (DAN MYER CARNIVAL)
3. You are observing a dolphin swim in the ocean when you notice a connection to your wonderful precalc class. No, you say, how can this be? Well you see this dolphin is diving in and out of the water
jumping up as high as 6 feet out of the water and going 6 feet below water before repeating these jumps.
Now, they say dolphins are really smart and this one (we’ll call Digger) really knows his trigonometry
and wants to somehow connect his jumps in and out of the water to the unit circle. Because of this
Digger repeats his jumps every 2 dolphin seconds. Sketch the graph of Digger swimming.
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For each of the following images, write a trigonometric function to describe the graph.
STEP 1: Identify what type of graph you are going to use and highlight one period.
STEP 2: Identify the characteristics of the graph (amplitude, period, phase shift and vertical shift; also
the sign of the lead coefficient.
STEP 3: Determine a, b, c and d
1.
3.
2.
4.
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Homework 4-8
5.
6.
7.
8.
9.
10.
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Date _________
Notes: 4-8 Continued
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use periodic functions to model and solve real world problems.
Vocabulary:
amplitude
period
beginning point
interval length
vertical translation
Example 1. Think back to the green dot on a wheel problem. – Diameter of 2 feet and 1 revolution
every 5 seconds…
Write the equation (assuming the green dot starts at the bottom of the ball):
What would the height of the green dot be at 3 seconds?
What would the height be at 12 seconds?
Example 2. Think back to the Ferris wheel problem. – Diameter of 50 feet and 1 revolution every 30
seconds…
Write the equation (assuming we load the bottom car 8 feet off the ground):
What would your height be at 6 seconds?
What would the height be at 20 seconds?
If you want to write this as a sine function, what would you have to do differently?
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Example 3. Think back to the dolphin problem. – jumping 6 feet out of the water and dives 6 feet
under, repeats jumps every 2 seconds…
Write the equation (assuming we begin to watch the dolphin as he is jumping out of the water):
What would the height be at 50 seconds?
What would the height be at  seconds?
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Homework 4-8
1. Think about our daily high temperatures throughout the year. The average temperature can be
modeled using a sinusoid. The lowest average high during the calendar year happens in January and is
28 degrees. The highest average high occurs in July and is 82 degrees. Sketch a graph (show 2
periods) and write an equation describing this situation.
2. You want to relive your glory days of being a kid and walked to a park to go a swing. You are sitting
their day-dreaming about bacon and chocolate and then notice that your swinging has created a
sinusoidal function. If the swing sits 2 feet off the ground and you can reach a height of 8 feet (going
forward and backward)! During the course of a minute, you go back and forth 10 times. Sketch a graph
showing your height above the ground as a function of time (assume your swinging starts at the bottom
of the swing). Write an equation to model this situation. (DAN MYER video clip)
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3. The Sun does some crazy things and for the past 200 years astronomers have kept track of the
number of sunspots that occur on the surface of the Sun and found that the number of sunspots can be
modeled using a sinusoid. The number of sunspots in a given year varies periodically, from a minimum
of about 10 a year to a maximum of about 110 per year. During these past 200 years there were exactly
10 cycles. Sketch a graph and write an equation of this situation. (The maximum number of sunspots
occurred on January 1, 2014, your graph should go forward from that point)
4. It is known that photosynthesis (the process by which plants convert carbon dioxide and water into
sugar and water) is a Circadian process exhibited by some plants. Assuming that a normal cycle takes
24 hours, the following is a typical graph of photosynthesis for a given plant. Here, photosynthesis is
measured in terms of carbon assimilation, the units of which are micromoles of carbon per m2/sec. Find
the equation to model this phenomenon.
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5. The motion of a weight on a certain kind of spring can be described by a modified trigonometric
function. At time 0, Carrie pushes the weight upward 3 inches from its equilibrium point and releases it.
The weight falls downward to a height 3 inches below its equilibrium point and the returns to the point
three inches above the equilibrium point after 2 seconds.
Sketch the height (distance from the equilibrium) of the weight as a function of time. Based on
your graph write an equation to describe this situation.
6. The graph below shows the deviation from the average daily temperature for the hours of a given
day, with t = 0 corresponding to 6 A.M. Use the graph to determine the related equation.
64
Date _________
Notes: 4-8 Continued
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use periodic functions to model and solve real world problems.
Vocabulary:
amplitude
period
beginning point
interval length
vertical translation
Use last night’s homework to answer these questions.
Use problem 1 (daily high temperatures throughout the year) to answer questions 1-3.
1. Predict the average temperature in April.
2. Is the location described in problem #1 in the northern or southern hemisphere. Explain how you
know.
3. Predict the average temperature in August.
65
Use problem 2 (your glory days on the swing) to answer questions 4-6.
4. Determine how far off the ground you are 27 seconds after you begin swinging.
5. Determine how far off the ground you are after 25.5 seconds after you begin swinging. Are you
going forward or backward (assume you started going backward)?
6. How much time will pass before you hit your maximum height the first time?
Use problem 3 (sunspots) to answer questions 7-9.
7. According to the model, how many sunspots will there be in 2017?
8. In 2044, how many sunspots should you expect?
9. How many sunspots will there be in 2041? Will there be more or less the following year?
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Use problem 4 (photosynthesis) to answer 10-12
10. After 7 hours, what is the rate of carbon assimilation?
11. After 7 hours, is the rate of carbon assimilation increasing or decreasing?
12. Rewrite this equation using a cosine function.
13. How would the equation you wrote for #5 (spring) change if you knew the equilibrium was 18
inches of the ground and y measured the distance off of the ground.
Use problem 6 to answer questions 14-15
14. Use the equation you found to predict the deviation in the temperature at 5PM.
15. If the average temperature for this day was 72◦, what was the temperature at midnight?
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Homework 4-8
16. Tidal Waves: Tsunamis, also known as tidal waves, are ocean waves produced by earthquakes
or other upheavals in the Earth’s crust and can move through the water undetected for hundreds
of miles at great speed. While traveling in the open ocean, these waves can be represented by a
sine graph with a very long wavelength (period) and a very small amplitude. Tsunami waves
only attain a monstrous size as they approach the shore, and represent a very different
phenomenon than the ocean swells created by heavy winds over an extended period of time.
A graph modeling a tsunami wave is given below. What is the height of the tsunami wave (from
crest to trough)? Note that h = 0 is considered the level of a calm ocean. What is the tsunami’s
wavelength? Find the equation for this wave.
17. A heavy wind is kicking up ocean swells approximately 10 feet high (from crest to trough), with
wavelengths of 250 feet. Find the equation that models these swells. Determine the height of a
wave measured 200 feet from the trough of the previous wave?
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Date _________
Notes: 4-8 Continue
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How are the six trigonometric and circular functions related to each other?
Learning Targets:
3. I can use periodic functions to model and solve real world problems.
Vocabulary:
Amplitude
period
beginning point
interval length
vertical translation
Example 1. As you ride a Ferris wheel, the height that you are above the ground varies periodically.
This Ferris wheel has a diameter of 38 feet and travels at a rate of 4 revolutions per minute. The car
clears the ground by 3 feet.
a. Sketch the height of the car on the Ferris wheel as a function of time.
b. Write a sine equation to describe the height of the seat as a function of time.
69
Example 2. Mark Twain sat on the deck of a river steamboat. As the paddle
wheel turned, a point on the paddle blade moved so that its distance y, in feet
from the water’s surface was a sinusoidal function of time t, in seconds. When
Twain’s stopwatch read 4 seconds, the point was at its highest, 16 ft above the
water’s surface. The wheel’s diameter was 18 ft, and it completed a revolution
every 10 seconds.
a. What is the lowest the point goes? Why is it reasonable for this value to be negative?
b. Find an equation for distance (in feet from the water’s surface) as a function of time. (HINT
– Make a sketch)
c. How far above the surface was the point when Mark’s stopwatch read 17 seconds?
d. What is the first positive value of t at which the point was at the water’s surface? At that
time was the point going into or coming out of the water? How can you tell?
70
Example 3. A chemotherapy treatment destroys red blood cells along with cancer cells. The red cell
count goes down for a while and then comes back up again. If a treatment is taken every three weeks,
then the red cell count resembles a periodic function of time (see the graph) If such a function is regular
enough, you can use a sinusoidal function as a mathematical model.
a. Suppose that a treatment is given at time 0 weeks, when
the count c=800 units. The count drops to a low of 200, and
then rises back to 800 when t=3, at which time the next
treatment is given. Write the equation for this sinusoidal
function.
b. The patient feels “good” if the count is above 700. Use the equation above to predict the
count 5 weeks after the treatment and thus conclude whether she has started feeling good again at
this time.
c. What about at 20 weeks?
d. For what interval of times around 9 weeks will the patient feel good? For how many days
does this good feeling last? Show (or explain) how you got your answers.
Example 4. The temperature in an office is controlled by an electronic thermostat. The temperatures

11 
vary according to the sinusoidal function: y  6sin  x 
  19 where y is the temperature (◦C) and
 12
12 
x is the time in hours past midnight.
a. What is the temperature in the office at 9AM when the employees come to work?
b. What are the maximum and minimum temperatures in the office?
71
Example 5. Zoey is at summer camp. One day she is swinging on a rope tied to a tree branch, going
back and forth alternately over the land and water. Nathan starts a stopwatch. At time x=2 seconds,
Zoey is at one end of her swing at a distance y = -23 feet from the river bank (see the figure). At time
x=5 seconds, she is at the other end of her swing at a distance y=17 feet from the riverbank. Assume
that while she is swinging y varies sinusoidally with x.
a. Write an equation to describe the distance from the riverbank. (Hint sketch graph first).
b. When x= 13 seconds, was Zoey over land or over water? Explain how you know.
Example 6. In predator-prey situations, the number of animals in each category tends to vary
periodically. A certain region has pumas as predators and deer as prey. The number of pumas varies
with time according to the function P  200sin(0.4t  0.8)  500 and the number of deer varies according to
the function D  400sin  0.4t   1500 with time t, in years.
a. How many pumas and deer will there be in the region in 15 years?
b. What is the maximum population of the pumas? When does this occur? What is the
minimum population of the pumas? When does this occur?
c. What is the maximum population of the deer? When does this occur? What is the minimum
population of the deer? When does this occur?
d. Using graphs of these two equations, make a statement regarding the relationship between
the number of puma and the number of deer.
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Homework 4-8
7. A satellite is deployed from a space shuttle into an orbit which goes alternately north and south of the
equator. Its distance from the equator over time can be approximated by a sine wave. It reaches
4500km, its farthest point north of the equator, 15 minutes after the launch. Half an orbit later, it is
4500km south of the equator, its farthest point south. One complete orbit takes 2 hours.
a. Find an equation of a sinusoidal function that models the distance of the satellite from the
equator.
b. How far away from the equator is the satellite 1 hour after launch?
8. Many animals exhibit a wavelike motion in their movements, as in the tail of a shark as it swims in a
straight line or the wingtips of a large bird in flight. Such movements can be modeled by a sine or
cosine function and will vary depending on the animal’s size, speed, and other factors. The graph below
models the position of a shark’s tail at time t, as measured to the left (negative) and right (positive) of a
straight line along its length. Use the graph to determine the related equation. Is the tail to the right,
left, or at center when t = 6.5 sec? How far? Would you say the shark is “swimming leisurely” or
“chasing its prey”? Justify your answer.
2. The typical voltage V supplied by an electrical outlet in the U.S. is a sinusoidal
function that oscillates between -165 volts and +165 volts with a frequency of 60
cycles per second. Obtain an equation for the voltage as a function of time t.
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4-8 Quiz Review
1. A team of biologists have discovered a new species in the rain forest. They note the temperature of
the animal appears to vary sinusoidally over time. A maximum temperature of 125◦ occurs 15 minutes
after they start their examination. A minimum temperature of 99◦ occurs 28 minutes later. The team
would like to find a way to predict the animal’s temperature over time in minutes. Your task is to help
them by creating a graph of one full period and an equation of temperature as a function over
time in minutes.
74
2. Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and forth across the
riverbank, going alternately over land and water. Jane decides to model his movement mathematically
and starts her stopwatch. Let t be the number of seconds the stopwatch reads and let y be the number of
meters Tarzan is from the riverbank. Assume that y varies sinusoidally with t and that y is positive
when Tarzan is over water and negative when he is over land.
Jane finds that when t = 4 Tarzan is at the end of his swing, where y = -40. She finds that when t = 9
he reaches the other end of the swing and y = 66.
a. Sketch a graph of this function.
b. Write an equation expressing Tarzan’s distance from the river bank in terms of t.
c. Find y when t = 5, t = 10, and t = 20.
d. Where was Tarzan when Jane started her stopwatch?
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3. A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to
time in days. He carefully collected data and graphed his resulting equation. From the graph,
determine the amplitude, period, phase shift and vertical shift of the graph. Write the equation of
the graph.
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4. An economist indicated that demand for temporary employment (measured in thousands of job
applications per week) in your county can be modeled by the function y  4.3sin 0.82(x  0.37  7.3
where x is time in years since January, 1995. Calculate the amplitude, the vertical shift, the phase
shift, and the period, and interpret the results (by filling in the blanks).
The demand for temporary employment has varied from a high of
thousand job applications over a
to a low of
year period.
 
5. The equation y  7sin  t  models the height of the tide along a certain coastal area, as compared to
6 
average sea level. Assuming t = 0 is midnight, predict the height of the tide at 5 A.M. Is the tide
rising or falling at this time?
77
6. The State Fish of Hawaii is the humuhumunukunukuapua’a (reef triggerfish), a small colorful fish
found abundantly in coastal waters. Suppose the tail motion of an adult fish is modeled by the equation
d t   sin15 t  with d(t) representing the position of the fish’s tail at time t (in seconds), as measured
in inches to the left (negative) or right (positive) of a straight line along its length. Is the tail to the left
or right of center at t=2.7 seconds? How far? Would you say this fish is “swimming leisurely” or
“running for cover”? Justify your answer.
7. Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 feet, and the loop spun
50 times each minute. If the lowest point on the rope was 6 inches above the ground, write an equation
to describe the height of this point above the ground after t seconds.
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8. A weight in a spring-mass systems exhibits harmonic motion that can be described by a sinusoid.
The system is in equilibrium when the weight is motionless. If the weight is pulled down or pushed up
and released it would tend to oscillate freely if there were no friction. In a certain spring-mass system,
the weight is 5 feet from a 10 foot ceiling when it is at rest (ie: the equilibrium point). The motion of the

weight can be described by the equation y  3sin   t   , where y is the distance from the equilibrium

2
point, and t is measured in seconds.
a. Find the period of this motion.
b. Find the amplitude of this motion.
c. Was the weight pulled down or pushed up before being released?
d. How far was the weight from the ceiling when it was released?
e. How far from the ceiling is the weight after 2.5 seconds?
f. How close will the weight come to the ceiling?
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g. When does the weight first pass its equilibrium point?
h. What is the greatest distance the weight will be from the ceiling?
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