Lesson Plan
Graphing Sine and Cosine
Lynn Weiss
Goal:
Students will use technology to discover how to graph trigonometric functions. They will work cooperatively to model sine and cosine functions after real world applications.
Prerequisites:
Students should have a basic knowledge of the unit circle. They should know how to graph
y=sin(x) and y=cos(x). They should understand basic trigonometric definitions such as amplitude, period, and shift.
Materials:
Students should have a TI Graphing Calculator (TI-82, TI-83, or TI-84).
Procedure:
In past years I have used a Spaghetti Lab where students work in groups to discover the graphs of y = sin(x) and y = cos(x). They use butcher paper and spaghetti along with their knowledge of right triangle trig and the unit circle to graph the functions. This lesson will begin on the school day following the completion of this lab.
Day 1:
1.)Review graphing y = sin(x) and y = cos(x). Be sure that students know the amplitude, period, domain and range of the functions. Discuss how to graph the functions in both radian and degree mode.
2.)Distribute the worksheet “Graphing Calculator Exploration, Graphing Sine and Cosine”. Have students work individually to complete the worksheet. They should discover how given the equation y = asinb(x-c)+d, changes in a, b, c, and d effect the graph.
3.)Go over the answers to the worksheet as a class. Have students volunteer to explain their solutions.
Day 2
1.)Give a pop partner quiz to assess students’ knowledge of graphing sine and cosine. The quiz will ask them to graph a function and to determine the equation of a given graph.
2.)Distribute index cards to all students in the class. Each classroom table will have a sign with an “answer” on it. The students will use their knowledge of the unit circle to find other members of their group. For instance a student with “sin 30◦” and a student with “cos 60◦” will each sit at the table with ½ as the answer. There will be a maximum of four students per group.
3.)Distribute the Ferris Wheel project to each student. Have them work with their group to complete the project.
Day 3
1.)Students will need additional time to work on the project. Collect the projects. Students will be assigned both a group and individual grade based on their effort in completing the project.
2.)Distribute the Ferris wheel extra credit assignment to those groups who desire a challenge.
Algebra Trig
Graphing Calculator Exploration
Graphing Sine and Cosine
Today we are going to use our graphing calculators to experiment with changing the values of a, b, c and d in the general equation:
y = a * sin (b(x - c)) + d.
Our goal is to discover how a, b, c, and d effect the graph.
First, use what you learned during the Spaghetti Lab, and graph y = sin(x)
and y = cos(x) from 0◦ to 360◦ below.
y = sin(x)
What are the values of a, b, c, and d?
a = b = c = d =
y = cos(x)
What are the values of a, b, c, and d?
a = b = c = d =
*Before we begin…
To what mode should our calculator be set?
Remember: The general equation is
y = a * sin (b(x - c)) + d
1.)Set your window to view x from 0 to 360 and y from -3 to 3.
Graph
As the absolute value of a increases, …
As the absolute value of a decreases,…
In the general form of the equation, changes in a impact the
______of the graph.
2.)Set your window to view x fro 0 to 720 and y from -2 to 2.
Graph
As b increases……
As b decreases…….
A change in b alters the ______of the graph of the equation.
3.)Set your window to view x from -90 to 540 and y from -2 to 2.
Graph
When c is positive(x - c), the graph ….
When c is negative(x + c), the graph….
A change in c causes a ______in the graph.
4.)Set your window to view x from 0 to 360 and y from -6 to 6.
Graph
When d is positive, the graph…
When d is negative, the graph…
A change in d causes a ______in the graph.
Now, use what you learned to complete the following.
1.)Determine the amplitude, period, phase shift, and vertical shift for each function:
Then, graph the function. Label the five critical points.
a.)
b.)
Write a sine and cosine equation for each given graph.
In general, when given a graph, how do you determine the values of a and b?
Algebra Trig.Good Luck to ______
Weiss
Pop Quiz on Sine and Cosine GraphsPeriod: ______
12 points – NO CALCULATOR
For #1 and #2, graph one full cycle. Be sure to label your axes and the five critical points.
1.
Amplitude = Period =
2.
Amplitude = Period =
For #3 and #4, write a sine and cosine equation to match the given graph.
3.)
4.)
Ferris Wheel Comparison Project
The 1893 Chicago World’s Fair is considered the birthplace of the classic amusement park ride, the Ferris wheel. The architectural wonder was created by an American engineer named George Ferris. The original Ferris wheel no longer exists. But, in 1990, a new Ferris wheel was built at Navy Pier in Chicago to resemble the original. While the Navy Pier Ferris Wheel is a beautiful Chicago landmark, its grandeur actually pales in comparison to Mr. Ferris’ creation.
The Ferris wheel built for the World’s Fair had a diameter of 250 feet. It stood 14 feet off the ground. It had 36 wooden boxcars that were the size of train cars. Each car could hold 60 people! The wheel would load cars in such a way that each rider could enjoy a full rotation that lasted about 10 minutes.
The Ferris wheel at Navy Pier has a diameter of 140 feet. It stands 10 feet off the ground. The wheel has 40 gondolas that seat six passengers each. It takes about 6 minutes for the Navy Pier Ferris Wheel to complete one rotation.
Below is a picture of the first Ferris wheel next to the Ferris wheel at Navy Pier.
In the space below is a diagram of the World’s Fair Ferris wheel and the boarding platform. Fill in the necessary information.
Now, complete the table below. Let h represent your vertical position (height) at time t where t is given in minutes. Remember, you need 5 critical t-axis values.
t h
Before you sketch the cosine and sine graphs, answer each of the following.
1.)What is the amplitude of the graph?______
a = ______
2.)What is the period of your curve? ______
(Remember )
b = ______
3.)What are your 5 critical t-values? ______
4.)What are the maximum and minimum values Max ______Min ______
of your curve? (Be careful!)
d = ______
5.)What is the height at time = 0 (y-intercept)? ______
Write a cosine equation for your curve.
(Remember there are an infinite number of possible answers)
______
Graph the equation.
Write a sine equation for your curve.
______
Graph the equation.
Now, answer the following questions. You may want to graph your equation on your graphing calculator.
1.)What is the circumference of the wheel?
2.)At what speed is the wheel traveling? Please give your answer in feet/second.
3.)If you begin your ride at the base of the wheel, what is your height after…
a.)1 minute?b.) 4 minutes?
4.)At what approximate time(s) will you reach the following heights?
a.) 100 ft.b.) 240 ft.
Now, repeat each step for the Navy Pier Ferris Wheel.
In the space below is a diagram of the Navy Pier Fair Ferris Wheel and the boarding platform. Fill in the necessary information.
Now, complete the table below. Let h represent your vertical position (height) at time t where t is given in minutes. Remember, you need 5 critical t-axis values.
t h
Before you sketch the cosine and sine graphs, answer each of the following.
1.)What is the amplitude of the graph?______
a = ______
2.)What is the period of your curve? ______
(Remember )
b = ______
3.)What are your 5 critical t-values? ______
4.)What are the maximum and minimum values Max ______Min ______
of your curve? (Be careful!)
d = ______
5.)What is the height at time = 0 (y-intercept)? ______
Write a cosine equation for your curve.
(Remember there are an infinite number of possible answers)
______
Graph the equation.
Write a sine equation for your curve.
______
Graph the equation.
Now, answer the following questions. You may want to graph your equation on your graphing calculator.
1.)What is the circumference of the wheel?
2.)At what speed is the wheel traveling? Please give your answer in feet/second.
3.)If you begin your ride at the base of the wheel, what is your height after…
- 1 minute?b.) 4 minutes?
4.)At what approximate time(s) will you reach the following heights?
a.) 100 ft.b.) 240 ft.
Ferris Wheel Extra Credit
1.)Imagine the Navy Pier and the Worlds Fair Ferris Wheel being built beside each other. If both wheels begin turning at once, over a 20 minute time period, at what times are the wheels at the same height?
2.)What is the length of the arc traveled by the Navy Pier Ferris wheel from the
4 o’clock to the 7 o’clock position?