module 7: trigonometric functions -...

Module 7: Trigonometric Functions

prepared by Susan Jones, Shelby High School Heather Raymond, Central Cabarrus High School

197 Sponsored by NC Math and Science Education Network

Chapter

7.1. North Carolina Standard Course of Study.......................................................................198

7.2. vocabulary......................................................198

7.3. introduction.................................................199

7.4. introductory material...................200

7.5. background mathematics......200

7.6. ties to textbooks .............................201

7.7. worked examples ...................................202

7.8. suggestions for homework.203

7.9. suggestions for projects.....203

7.10. introductory material...................204

7.11. background mathematics......204

7.12. ties to textbooks...............................204

7.13 worked example .......................................205

7.14 suggestions for homework.206

7.15 suggestions for projects.....206

7.16 biorhythms & sine curves .......207

7


7.1 North Carolina Standard Course of Study

GOAL 2: The learner will use functions to solve problems.

2.04 Use trigonometric (sine, cosine) functions to model and solve problems; justify results.

a) Solve using tables, graphs, and algebraic properties. b) Create and identify transformations with respect to period, amplitude, and vertical

and horizontal shifts. c) Develop and use the Law of Sines and the Law of Cosines.

7.2 Vocabulary

Graph Independent Dependent Domain/Range Period Amplitude Phase Shift Frequency Coefficients

sin( )y a bx c d= + +

+

cos( )y a bx c d= + Intercepts Law of Sines Law of Cosines Unit Circle Radian/Degree Measure Special Angles (multiples of π, π/2, π/3, π/4,

π/6) Solve Equations, Justifying Steps Used

Module 7: Trigonometric Functions 198


7.3. Introduction

The study of trigonometry can be loosely categorized into two sections, trigonometry as a study of triangles and trigonometry as a study of circular functions. Although related, these two topics can be taught individually. Whatever your choice, be sure to stress the relationship between the unit circle relationships for sine and cosine and the right triangle relationships.

In the Algebra and Trigonometry textbook, Chapter Seven includes triangle trigonometry and Chapter Eight includes circular functions and sinusoids. These two chapters are written so that they can be taught in any order. In the Functions Modeling Change textbook, Chapter Six introduces trigonometric functions as circular functions. This follows the chapter on transforming graphs and flows very well into a study of sinusoidal functions. The tools for Chapter Six include triangle trigonometry. Chapter Six, the tools for Chapter Six, and the first section of Chapter Seven would cover the material necessary for the trigonometry module for this course.

Algebra and Trigonometry provides a more balanced look at both triangle trigonometry and circular trigonometry. If you are using this textbook, you can decide what your main focus is to be and pick and choose accordingly. Functions Modeling Change puts greater emphasis on using trigonometric functions to model periodic behavior and much less on triangle trigonometry. If you are using this textbook, it would seem very logical to do the same thing.


Module 7: Trigonometric Functions Triangle Trigonometry

7.4. Introductory material

Measuring the height of the school flag pole

You will need a flagpole, or other vertical distance that cannot be directly measured, a long measuring tape, and an angle measuring device.

1) Measure a distance from the base of the flag pole.

2) Measure the angle to the top of the flag pole.

3) Measure the angle to the bottom of the flag pole.

4) Record all measurements.

5) Discuss and obtain any other information you might need (height of person making angle measurements) or assumptions that you will need to make (flag pole and person are vertical and distance from base is horizontal.)

6) Using knowledge that students should have from geometry, label as many angles and lengths as possible on the diagram from the data collected. This will be a good way to review some basic concepts about triangles and parallel lines.

This problem can be finished using several different trigonometric concepts that will be taught in this module. As you teach each concept, return to this problem and calculate the flag pole height.

7.5. Background mathematics

Even though some of this material is more geared to the trigonometry of circular functions, it is still good material to explain in the beginning of the trigonometry module. It is necessary introductory material for trigonometry no matter if the focus will be first on triangles or circular functions and sinusoids.

• Definition of an angle as a rotation about a point • Radian measure of angles, conversions to degrees/radians • Angles in standard position • Coterminal angles • Reference angles • Arc length and area of a sector of a circle • Right triangles: define the six trigonometric functions using the right triangle (mention

secant, cosecant, and cotangent but main focus will be using sine, cosine, and tangent), solving triangles

• Area: 1 sin2

A ab C= using 2 lengths and the included angle

• The Law of Sines and Law of Cosines



• Angle of elevation or inclination, angle of depression, line of sight

7.6. Ties to sections in textbooks

Algebra and Trigonometry Chapter 7:

• Section 7.1 • Section 7.2: note: you may choose to omit the section on special triangles (special angles)

for work with the unit circle and circular functions • Section 7.3: omit trig functions secant, cosecant, and cotangent, omit all trigonometric

identities • Section 7.4 • Section 7.5: omit Heron’s formula

NOTE: main focus of this part of the module is solving triangles and applications of that. Some of the material included here can be mentioned quickly. Spend the majority of your time and energy on solving triangles, finding area of triangles, and working application problems involving these concepts.

Functions Modeling Change The trigonometry in this text book puts more emphasis on trigonometric functions to model periodic behavior. If you are teaching from this book and want to include more triangle trigonometry, you will need to supplement the material included and teach it after you have completed the material in chapter six. However, you may find that following the format of this book covers all trigonometry topics adequately.

• Tools for Chapter six • Chapter 7: Section 7.1



7.7. Worked Example

PROBLEM: Acme Oil Company needs to drill a new oil well in the Gulf of Mexico just off the coast of Louisiana at point C. There is an existing well at point B. Regulations state that there must be at least 2000 feet between oil wells. Acme Surveying is hired to check the distance between the existing well and the proposed new well. The surveyors set up points A and D on the shore and take the following measurements with a total station:

The distance AD is 3779.56 ft 55 22 '04"BAD= ° 47 27 '30"ADB= °33 00 '53"CAD= ° 91 32 '08"ADC = °

What is the distance between the well at point C and the drilling site at point B?

B

A D

C

Solution: BC=2228.50 ft

WORK: Label the intersection of BD and AC point E. Find as many angles for the diagram as you can:

55 22 '04"BAD= ° 47 27 '30"ADB= ° 33 00 '53"CAD= ° 91 32 '08"ADC = ° 44 04 '38"BDC = ° 22 21'11"BAC = ° 55 26 '59"ACD= ° 77 10 '26"ABD= ° 99 31'37"BEC = °

80 28'23"CED AEB= = °

Using the Law of Sines find distances:

sin 99 31'37" sin 47 27 '30" 2823.673779.56

AEAE

° °= =

sin 22 21'11" sin 77 10 '26" 1101.362823.67

BEBE° °= =



sin 33 00 '53" sin 99 31'37" 2088.113779.56

AEDE° °= =

sin 55 26 '59" sin 44 04 '38" 1763.592088.11

CECE

° °= =

Using the Law of Cosines: find BC

( )( )2 2 21101.36 1763.59 2 1101.36 1763.59 cos99 31'37"2228.50

BCBC= + − °=

7.8. Suggestions for homework

The problems listed are problems that are appropriate for the course. You may choose to select a few of them for your students, give only odd or even problems, or give the entire problem set, whatever is appropriate for the students you teach.

Algebra and Trigonometry Chapter 7

• Section 7.1: 1-63 • Section 7.2: 1-3 (find only sine, cosine, and tangent values), 5, 6, 11-14, 15, 16, 18 (only

work with sine, cosine, and tangent), 28-30, 33-47, 49-51 (If you have included special angles, students can do 1-51.)

• Section 7.3: 1-6, 31, 32, 41-43, 47, 48, 50-54 (If you have included special angles, students can do 1-34, and 50-54.)

• Section 7.4: 1-31 • Section 7-5: 1-36, 39-42

Functions Modeling Change

• Tools for Chapter six: 1-29 • Section 7.1: 1-37

7.9. Suggestions for Projects

1. Wake Technical Community College Timber Calculation Activity

2. Trig-Star Contest Trig-Star is a national trigonometry contest sponsored by the National Society of Professional Surveyors. It is open to all high school students. For more information, contact Don Wilson (704) 482-4145.


Module 7: Trigonometric Functions Trigonometry of Circular Functions

7.10. Introductory material

PROBLEM:

Suppose a mass suspended on a spring is bouncing up and down. Assume perfect elasticity and no air resistance. The object’s distance from the floor when it is a rest is 1m. The maximum displacement is 10cm as it bounces. It takes 2 seconds to complete one bounce or cycle. Suppose the mass is at rest at t=0 and that the spring bounces up first.

1. What will the distance from the floor be at t=.5sec, t=1sec, t=1.5sec, and t=2sec? 2. What will the distances be for the second bounce? The third bounce?

Set up a table. Notice that the distances begin to repeat themselves at t=2sec.

3. Plot this data on a graph. How should you connect these data points to model the object’s distance from the floor as it bounces? Should you use line segments or a curve to connect the data points? Why?

Use this problem to introduce the concepts of periodic functions, amplitude, and midline. As you teach graphing sine and cosine functions, come back to this example and have students write an actual equation to model this situation.

7.11. Background Discussion of Mathematics

• Definition of an angle as a rotation about a point • Radian measure of angles, conversions to degrees/radians • Angles in standard position • Coterminal angles • Unit Circle • Sine and cosine functions defined for a unit circle and a circle of radius r • Graphs of sine and cosine functions • Sinusoidal functions: period, amplitude, maximum value, minimum value, midline, phase

shift or horizontal shift, vertical shift

7.12. Ties to sections in textbooks


• Section 8.1 • Section 8.2: omit trigonometric identities; mention secant, cosecant, and cotangent

functions but emphasize sine, cosine, and tangent functions • Sections 8.3: work only with sine and cosine functions in the form and

Functions Modeling Change Chapter 6:

• Section 6.1 • Section 6.2



• Section 6.3 • Section 6.4 • Section 6.5 • Section 6.7: visit solving trigonometric equations and the inverse sine and cosine

functions briefly so that students will be able to do necessary calculations with the Law of Sines and Cosines when you get to that. You may wish to wait until you introduce those laws in 7.1.

7.13. Worked Example

Modeling periodic data

( )sin ( )y A B t h k= − +

The table below lists the normal average high temperatures for Charlotte, NC.

Month Month Temp °F Month Month Temp °F January 1 51.3 July 7 90.1 February 2 55.9 August 8 88.4 March 3 64.1 September 9 82.3 April 4 72.8 October 10 72.6 May 5 79.7 November 11 62.8 June 6 86 December 12 54

(Data collected from www.noaa.gov/climate.htm)

1. Create a scatter plot of the data. Sketch your plot.

2. What is the period of the function? What is the value of B for the periodic function?

3. What is the minimum temperature? What is the maximum temperature? Use these values to find the amplitude and midline. What are the values of A and k for the periodic function?

4. Put the value of k in your calculator to graph on your data set. Trace along this line to find the x-value where the data set would cross the midline. You will need to estimate this value. What is the value of h for the periodic function?

5. Put your values of A, B, h, and k into the equation .

6. Graph this equation on your data. How well does it fit?

SOLUTION:

2. Period=12 months 6

B π=


http://www.noaa.gov/climate.htm


3. Min=51.3 Max=90.1 90.1 51.3amplitude 19.4 19.4

2A−= = =

midline 90.1 19.4 70.7 70.7k= − = =

4. 3.9x h≈ =

5. 19.4sin ( 3.9) 70.76

y tπ⎛ ⎞⎟⎜= − +⎟⎜ ⎟⎜⎝ ⎠

7.14. Suggestions for homework

The problems listed are problems that area appropriate for the course. You may choose to select a few of them for your students, give only odd or even problems, or give the entire problem set, whatever is appropriate for the students you teach.


• Section 8.1: 1-44 • Section 8.2: 1-7, 10-13, 21-24( sint, cost, and tant only), 25-32, 45,46,59-61, 64, 66 • Section 8.3: 1-8, 11-38

Functions Modeling Change Chapter 6:

• Section 6.1: 1-31 • Section 6.2: 1-31, 34-36 • Section 6.3: 1-34, 36-40 • Section 6.4: 1-25 • Section 6.5: 1-21, 24-29, 31, 32, 38, 40

Suggestions for projects

1. Algebra and Trigonometry Discovery Project p.540 2. Wake Technical Community College Modeling Periodic Data Activity 3. National Oceanic and Atmospheric Administration website may have additional data you

could use for modeling periodic behavior. (www.noaa.gov/)


http://www.noaa.gov/


Biorhythms and Sine Curves from SAS

Overview In this project, students compute and graph their biorhythms. The term biorhythms refers to the idea that each person has three biological cycles occurring in periodic patterns, like sine curves. Students compare their three cycles by graphing their "physical," "emotional," and "intellectual" biorhythms for the current month.

Purpose To practice graphing circular functions

Duration About one hour

Objectives The student will

• Determine the graphs of his or her three biorhythm curves • Plot these curves for the current month

Materials and Resources Graph paper Colored pencils External web sites: Daily Biorhythms (www.bio-chart.com) Biorhythms on the Web (www.arakni.com/biorhythm)

Procedures 1. If a person's lifetime biorhythms were graphed, they would form a continuous sine curve. To

discover and then graph where each student currently is along this curve, students first calculate how many days they have been alive, up to the first day of this month. Then for each biorhythm, they compute the number of complete cycles thus far, by dividing the total days of their lifetime by the number of days in each cycle. The physical cycle is 23 days, the emotional cycle is 28 days, and the intellectual cycle is 33 days.

2. In most cases, the number of complete cycles will result in whole number plus a decimal. To determine the starting point of the graph, students multiply the fractional part of the number (rounded to the thousandths) of complete cycles by the total number of days in the cycle. The answer obtained from this calculation indicates where they were in a particular cycle on the first day of the month.

3. The value of the sine curve is at zero at the beginning of its cycle, halfway through its cycle and at the end of its cycle. It reaches its maximum at one-fourth of the total cycle and its minimum at three-fourths. For each of the three biorhythm cycles, students determine on what day each of these special points occur. These five points divide the cycle into four equal intervals. As a result, every biorhythm cycle will need to be divided by four. For example, the physical cycle's twenty-three days divided by four is 5.75, therefore the special points occur at 0, 5.75, 11.5, 17.25 and 23 days. Students sketch each of the three cycles on a



separate graph using the correct period and locating these special points. They may use any convenient amplitude.

4. Using the numerical answers from step 2, students locate each value on the x-axes of the appropriate graph. This is the location for the beginning of the graph representing the first day of this month.

5. On another sheet of graph paper, students graph all three cycles for the current month on the same axes, using a different color for each graph and labeling appropriately. Each of the graphs will display different periods.

6. After graphing their cycles, students check their results with those given at the suggested web sites. Biorhythms on the Web displays the cycles in a graph similar to the one the students create.

Conclusion According to their biorhythm scales, when would it be the best day to take an especially difficult exam? When would it be the best to run a marathon? When would it be best to confront something that is emotionally draining? Do students agree or disagree with this "theory" of biorhythms?

Figure 1


module 7: trigonometric functions -...

Documents