Contents How You Ma Use This Resource Guide ii 0 Trigonometric Formulas, Identities, and Equations Worksheet 0.: Graphical Analsis of Trig Identities.............. Worksheet 0.: Verifing Trigonometric Identities: A Graphical Approach.. Worksheet 0.: Solving Trigonometric Equations: A Graphical Approach... 7 Worksheet 0.: Matching: A Summar Eercise of Trig Formulas....... 9 Worksheet 0.5: Modeling and Predicting.................... 0 Answers i
How You Ma Use This Resource Guide This guide is divided into chapters that match the chapters in the third editions of Technical Mathematics and Technical Mathematics with Calculus b John C. Peterson. The guide was originall developed for the second editions of these books b Robert Kimball, Lisa Morgan Hodge, and James A. Martin all of Wake Technical Communit College, Raleigh, North Carolina. It has been modified for the third editions b the author. Each chapter in this Resource Guide contains the objectives for that chapter, some teaching hints, guidelines based on NCTM and AMATYC standards, and activities. The teaching hints are often linked to activities in the Resource Guide, but also include comments concerning the appropriate use of technolog and options regarding pedagogical strategies that ma be implemented. The guidelines provide comments from the Crossroads of the American Mathematical Association of Two-Year Colleges (AMATYC), and the Standards of the National Council of Teachers of Mathematics, as well as other important sources. These guidelines concern both content and pedagog and are meant to help ou consider how ou will present the material to our students. The instructor must consider a multitude of factors in devising classroom strategies for a particular group of students. We all know that students learn better when the are activel involved in the learning process and know where what the are learning is used. We all sa that less lecture is better than more lecture, but each one of us must decide on what works best for us as well as our students. The activities provided in the resource guide are intended to supplement the ecellent problems found in the tet. Some activities can be quickl used in class and some ma be assigned over an etended period to groups of students. Man of the activities built around spreadsheets can be done just as well with programmable graphing calculators; but we think that students should learn to use the spreadsheet as a mathematical tool. There are obstacles to be overcome if we are to embrace this useful technolog for use in our courses, but it is worth the effort to provide meaningful eperiences with spreadsheets to people who probabl will have to use them on the job. Whether or not ou use an of the activities, we hope that this guide provides ou with some thought-provoking discussion that will lead to better teaching and qualit learning. ii
Chapter 0 Trigonometric Formulas, Identities, and Equations Objectives After completing this chapter, the student will be able to: Verif identities using the basic eight identities; Use sum, difference, double, and half angle formulas to simplif epressions and verif identities; Solve trigonometric equations using trigonometric identities. Teaching Hints. Emphasize using trig identities to simplif epressions rather than finding eact values of trigonometric functions. To successfull solve trig equations, one needs to first know how to simplif epressions and rearrange the form of an equation.. Show noneamples as well as eamples of identities. Graphicall and numericall verif equations that are identities and show eamples of commonl mistaken identities. For eample, show sin() does not equal sin() b having students evaluate both for various values of and b graphing both of them on the same screen to show the inequalit. (refer to Activit 0.). Prove the identit for the sin(a+b) or cos(a+b) formula geometricall. Students do not see or perform ver man formal proofs in this course. The geometrical proofs of the sum formulas provide a review of right-triangle trig and visuall clarif the sum of the two angles A and B, which gives students a better understanding of the sum formulas.. After students have studied and learned the identit formulas individuall, give them eperiences in choosing which formula applies when all are mied into the eercises. (refer to Activit 0.) 5. Illustrate solving trigonometric equations with applications. For eample, in Chapter 0, circular motion was modeled b sines and cosines. Review modeling circular
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition motion and appl solving trigonometric equations to predicting times for specific locations of the circular motion. (refer to Activit 0.) Guidelines This chapter is an etension of Chapter 0. It provides ou the opportunit to spiral through modeling periodic behavior again, onl now ou can algebraicall support the predictions made based on a model b appling the trig identities to solve trig equations. However, emphasis should not be on the algebraic solutions, since the Crossroads document recommends: increased attention given to determining real roots of an equation b a combination of graphical and numerical methods and decreased attention... given to reduction formulas and proofs of complicated trig identities. Algebraic support of solving trig equations should be required (ecept ver complicated ones), but the graphical and numerical methods of solving should also be included and used on the ver complicated equations. Also, the Crossroads document calls for the instructor to introduce the different concepts and techniques of solving equations in the contet of solving real problems. Modeling periodic phenomena is a realistic problem and should be dealt with in this chapter. Guidelines for Content (NCTM Standards) Increased Attention Realistic applications and modeling Connections among right triangles, trig functions, and circular functions Use of graphing utilities to solve equations and inequalities Functions constructed as models of realworld problems Decreased Attention Verification of comple identities Numerical application of trigonometric formulas Pencil-and-paper solutions of trig equations Formulas given as models of real-world problems Activities. Graphical Analsis of Trig Identities. Verifing Trigonometric Identities: A Graphical Approach. Solving Trigonometric Equations: A Graphical Approach. Matching: A Summar Eercise of Trig Formulas 5. Modeling and Predicting
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition Student Worksheet 0. Graphical Analsis of Trig Identities. Graph each set of epressions to determine an identit for that row. Also, make note of epressions that are not alwas equal. (a) sin( + ) sin() + sin() sin() cos() + sin() cos() (b) cos( ) cos() cos() cos() cos() + sin() cos() (c) sin() sin() sin() cos() (d) sin() sin() sin() cos() sin() cos() (e) cos() cos() cos () sin () cos () + sin () cos() cos() (f) sin() cos() + cos() + cos() (g) sin() cos(). Write the seven identities demonstrated above. (a) sin( A + B ) = sin( A ) cos( B ) + sin( B ) cos( A ) (b) (c) (d) (e) (f) (g). Write seven warnings or cautions about misuses of identities. (a) sin( A + B ) sin( A ) + sin( B ) (b) (c) (d) (e) (f) (g). Write a short paragraph describing how a graphing utilit can be used to verif which equations are identities and which are not.
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition Student Worksheet 0. Verifing Trigonometric Identities: A Graphical Approach A trigonometric identit is a statement of equalit that is true for all angles for which the trigonometric ratios in the identit are defined. In this activit ou will use a graphing utilit to verif some of the basic identities. Eercises. Use a graphing calculator with the following window settings to graph the following trigonometric functions and sketch the graph on the ais provided. Min Ma Scl 8 Min Ma Scl (a) = sin cos (b) = cos sin (c) = + cos (d) = sin cos (e) = sec (f) = csc (g) = cos + sin (h) = cos sin (i) = cot
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 5 (j) = + tan (k) = cot (l) = cos ( ) (m) = cos() (n) = tan (o) = sin() (p) = + cot (q) = cos (r) = (s) = sin (t) = sin () + cos () (u) = tan Comparing graphs (a) (u) ou will notice that some of the curves are identical. For eample, the graph in (a) is identical to the graph in (n) for all angles for which the tan is defined. This verifies the identit: sin cos = tan. Complete the table at the top of the net page to verif other identities.
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 6 Identical Graphs (a), (k), and (n) Identities Verified sin cos = tan = cot s
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 7 Student Worksheet 0. Solving Trigonometric Equations: A Graphical Approach In this activit ou will use a graphing utilit to solve trigonometric equations. I. Use a graphing calculator with the following window settings to solve the following trigonometric equations for 0 <. Min 6 Ma 6 Scl 6 Min Ma Scl. cos sin = [Graph: = cos sin ]. tan = [Graph: = tan ] Solutions: Solutions:. cos() = sin. sec sec = Solutions: 5. cot = tan Solutions: 6. sin ( ) = sin ( ) Solutions: Solutions:
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 8 7. tan = sin tan 8. sin = cos ( ) Solutions: Solutions: II. Solve the following application problems. Set the window settings on our calculator according to the needs of the problem. 9. In Boston the number of hours of dalight D(t) on a particular da of the ear ma be approimated b ( ) D(t) = sin (t 79) + 65 with t in das and t = 0 corresponding to Januar. How man das of the ear have more than.5 hours of dalight? 0. A tidal wave of height 50 feet and period 0 minutes is approaching a sea wall that is.5 feet above sea level. From a particular point on shore, the distance from sea level to the top of the wave is given b ( ) t = 5 cos 5 with t in minutes. For approimatel how man minutes of each 0-minute period is the top of the wave above the sea wall?
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 9 Student Worksheet 0. Matching: A Summar Eercise of Trig Formulas Simplif each trig epression in the left-hand column and match with an equivalent epression pn the right-hand column.. sin() cos(5) cos() sin(5). sin () cos (). sin() cos(). cos () + sin () 5. cos() cos() + sin() sin() 6. sin (6) cos() 7. cos 8. sin 9. cos () 0. + cos(). 8 cos (6) (a) sin (b) sin() (c) sin() (d) sin (e) sin () (f) sin(6) (g) cos() (h) cos(6) (i) cos(6) (j) cos() (k) cos() (l) cos(). + cos() (m) cos() (n) cos() (o) tan (p) cot (q) (r) (s)
Instructional Resource Guide, Chapter 0 Peterson, Technical Mathematics, rd edition 0 Student Worksheet 0.5 Modeling and Predicting For each application below, find a mathematical model and use solving techniques to make predictions based on our model.. A Ferris wheel of radius feet rotates one time ever 0 seconds. (a) Assuming the Ferris wheel begins to rotate when ou get on the wheel at ground level (i.e., at t = 0 our height is 0 feet and when t = 0 our height is 6 feet), write our height as a function of time in seconds. (b) Algebraicall and graphicall, predict the times at which ou will be at a height of 50 feet above the ground. (c) Algebraicall and graphicall show that there are no times at which our height will be 70 feet.. The man on the fling trapeze shown at the right takes 5 seconds to leave and return to the platform once. (a) Assuming the swing has a radius of 50 feet and the platform is 80 feet above the ground, write functions for the height of the trapeze artist above the ground and the horizontal distance from the platform as functions of time. (b) Algebraicall, predict the times at which the height of the man on the fling trapeze is 65 feet. (c) Algebraicall and graphicall predict the times at which the horizontal distance from the platform is 0 feet. 80 feet 50 feet (d) Algebraicall and graphicall determine if the trapeze artist will be at a height of 60 feet at the same time his horizontal distance from the platform is 0 feet.. The data below show the temperatures for a certain region for one ear specificall, the date, number of das after Januar, and the average temperature in degrees Fahrenheit for that da. (a) Write a sine function that approimatel models the average temperatures of this region during the given ear. For approimating purposes, use (, 78) and (0, 6) as (da, temp) when the maimum and minimum average temperatures for the ear. (b) Use this model to algebraicall and graphicall predict the dates of the following ear when the temperature will reach 70 degrees. (c) If ou run our air conditioner on das when the temperature is greater than 70 degrees, how man das during the ear do ou epect to run our AC? (d) How man das of the ear will ou run our heating unit if it runs on das when the temperature is below 5 degrees? Date /0 /0 /0 /8 /0 /0 /0 /0 5/0 5/0 6/0 6/0 Da of Year 0 0 59 69 89 00 0 0 50 6 8 Avg. Temp 9 6 9 50 57 6 65 70 7 77 ( F) Date 7/0 7/0 8/0 8/0 9/0 9/0 0/0 0/0 /0 /0 /0 /0 Da of Year 9 5 7 8 0 6 Avg. Temp 79 78 75 7 7 65 59 5 8 0 7 ( F)
Answers Student Worksheet 0.. (a) = sin( + ) = sin() + sin() = sin() cos()+sin() cos() (b) = cos( ) = cos() cos() = cos() cos()+sin() sin() (c) = sin() = sin() = sin() cos() (d) = sin() = sin()
Instructional Resource Guide, Answers Peterson, Technical Mathematics, rd edition = sin() cos() = sin() cos() (e) = cos() = cos() = cos () sin () = cos () + sin () (f) = cos() == cos() = sin() (g) + cos() = b. cos(a B) = cos A cos B + sin A sin B c. sin(a) = sin A cos A d. sin(a) = sin(a) cos(a) == + cos() = cos()
Instructional Resource Guide, Answers Peterson, Technical Mathematics, rd edition e. cos(a) = cos (A) sin (A) cos(a) f. sin(a) = ± + cos(a) g. cos(a) = ± b. cos(a B) cos A cos B c. sin(a) sin A d. sin(a) sin(a) and sin(a) sin A cos A e. cos(a) cos(a) and cos(a) cos (A) + sin (A) cos(a) cos(a) f. sin(a) and sin(a) + cos(a) + cos(a) g. cos(a) and cos(a) Student Worksheet 0.. Identical Graphs (a), (k), and (n) (b), (i), and (u) (c) and (l) (d) and (o) (e) and (j) (f) and (p) (g), (r) and (t) (h), (m), (q), and (s) Identities Verified sin cos = tan = cot cos sin = cot = tan + cos = cos sin cos = sin() sec = + tan csc = + cot cos + sin = = sin () + cos () cos sin = cos() = cos = sin Student Worksheet 0. I. 6, 5 6, and.5 I. 6, 5 6, 7 6, and 6, I. 0.5 and.5 I. 0,, and 5 I5.,,, and 5 I6.,, and 5
Instructional Resource Guide, Answers Peterson, Technical Mathematics, rd edition I7. 0,,,,, and 5 I8. II9. about das II0. 0 minutes Student Worksheet 0. 0.. (c). (h). (f). (q) 5. (g) 6. (l) 7. (a) 8. (o) 9. (e) 0. (n). (j). (m) Student Worksheet 0.5 a. h(t) = cos ( ) t + 0 b. Algebraicall: ( ) t cos + = 50 0 ( ) t cos = 8 Graphicall: cos c. Algebraicall: 0 ) ( t 0 t 0 = 8 = 9 6 ( = cos 9 ) 6 ( 9 ) 6 6.906 and.090 t = 0 cos ( ) t cos + = 70 0 ( ) t cos = 8 cos 0 ) ( t 0 = 8 = 9 6 Since the cosine function has a range of [, ] there are no values of t that will satisf this solution. Graphicall: From the figure below ou can see that the graph for the height of the Ferris wheel never reaches the line = 70. Thus, there are no values of t that will satisf this problem. ( ) a. Height: (t) = 80 t 50 sin, Horizontal dis- 5 ( ) tance from the platform: (t) = t 50 sin, where t is 0 in seconds and and are in feet. b. ( ) 80 t 50 sin = 65 5 ( ) t 50 sin = 5 5 ( ) t sin = 0. 5 t = 5 sin 0. 0.85,.68, 5.85, and 9.68 c. Algebraicall: ( ) t 50 sin = 0 0 ( ) t sin = 0.8 0 t = sin 0.8 0 t = 0 sin 0.8.95, and 7.08 Graphicall:
Instructional Resource Guide, Answers Peterson, Technical Mathematics, rd edition 5 d. Algebraicall: We first find the time when the horizontal distance is 0 feet and then substitute this value in the height function to see if we get 60 ft. ( ) t 50 sin = 0 0 ( ) t sin = 0. 0 t 0 = sin 0. t = 0 sin 0. Substituting this value of t in the height function we get ( ) ( 0 ( 0 sin 0. = 80 50 sin sin 0. ) ) b. about Ma 6 5 = 80 ( 50 sin sin 0. ) c. about 08 das. Thus, we see that at the time the trapeze artist is 0 ft awa from the platform he is about feet above the ground. Graphicall: In the figure below, the horizontal line is = 0. As ou can see from the graphs, at the time that this intersects the function for the horizontal distance, the trapeze artist s height is about.5 ft above the ground. a. F (d) 0.7 sin(0.075d.95) + 57.7 F on da d of the ear. d. about 0 das