Applications of Trigonometric and Circular Functions

Transcription

1 CHAPTER OBJECTIVES Applications of Trigonometric and Circular Functions Stresses in the earth compress rock formations and cause them to buckle into sinusoidal shapes. It is important for geologists to be able to predict the depth of a rock formation at a given point. Such information can be ver useful for structural engineers as well. In this chapter ou ll learn about the circular functions, which are closel related to the trigonometric functions. Geologists and engineers use these functions as mathematical models to perform calculations for such wav rock formations. 8 Learn the meanings of amplitude, period, phase displacement, and ccle of a sinusoidal graph. Given an one of these sets of information about a sinusoid, find the other two: the equation; the graph; the amplitude, period or frequenc, phase displacement, and sinusoidal ais. Plot the graphs of the tangent, cotangent, secant, and cosecant functions, showing their behavior when the function value is undefined. Given an angle measure in degrees, convert it to radians, and vice versa. Given an angle measure in radians, find trigonometric function values. Learn about the circular functions and their relationships to trigonometric functions. Given the equation for a circular or trigonometric function and a particular value of, find specified values of or graphicall, numericall, and algebraicall. Given a verbal description of a periodic phenomenon, write an equation using the sine or cosine function and use the equation as a mathematical model to make predictions and interpretations about the real world. Given information about a rotating object or connected rotating objects, find linear and angular velocities of points on the objects. 8

2 Chapter 6 Applications of Trigonometric and Circular Functions Overview In this chapter, students learn to sketch a graph of an sinusoid from its equation. B reversing the graph-sketch process, students learn to write an equation for a sinusoid with an given period, amplitude, phase displacement, or vertical position. Thus, the can use sinusoidal functions as mathematical models of periodic functions in the real world. Intro ducing circular functions with arguments in radians makes trigonometric functions more useful in applications where the independent variable is time or distance rather than angle. Students then appl their knowledge of angles and circular functions to find angular and linear velocities in rotar motion. Using This Chapter This chapter epands on Chapter 5 with graphs of all si trigonometric functions and their transformations. Upon completing this chapter, students should be able to use degrees and radians interchangeabl depending on the contet of the problem. Students will also have the the tools to create and use mathematical models from verbal descriptions of a periodic phenomenon to make predictions and interpretations about realworld situations. The section on rotar motion ma be omitted. Teaching Resources Eplorations Eploration 6-a: Sine and Cosine Graphs, Manuall Eploration 6-: Periodic Dail Temperatures Eploration 6-a: Transformed Sinusoid Graphs Eploration 6-b: Sinusoidal Equations from Graphs Eploration 6-a: Tangent and Secant Graphs Eploration 6-b: Transformed Tangent and Secant Graphs Eploration 6-4: Introduction to Radians Eploration 6-4a: Radian Measure of Angles Eploration 6-5a: Circular Function Parent Graphs Eploration 6-6a: Sinusoids, Given, Find Numericall Eploration 6-6b: Given, Find Algebraicall Eploration 6-7: Chemotherap Problem Eploration 6-7a: Oil Well Problem Eploration 6-8: Angular and Linear Velocit Eploration 6-8a: Adam Ant Problem Eploration 6-8b: Motorccle Problem Eploration 6-9a: Carbon Dioide Follow-Up Eploration 6-9b: Rehearsal for Sinusoids Test Blackline Masters Sections 6-, 6-4, and 6-9 Supplementar Problems Sections 6-, and 6-5 to 6-7 Assessment Resources Test 5, Sections 6- to 6-, Forms A and B Test 6, Sections 6-4 to 6-8, Forms A and B Test 7, Chapter 6, Forms A and B Technolog Resources Dnamic Precalculus Eplorations Variation of Tangent and Secant Sinusoid Translation Sinusoid Dilation The Inequalit sin tan Waterwheel Sketchpad Presentation Sketches Trig Tracers Present.gsp Radians Present.gsp Circular Functions Present.gsp Circular Transforms Present.gsp Sine Challenge Present.gsp Activities Sketchpad: Trigonometr Tracers Sketchpad: Transformations of Circular Functions CAS Activit 6-4a: Inverse Trigonometric Functions CAS Activit 6-7a: Epicenter of an Earthquake 8A Chapter 6 Interleaf: Applications of Trigonometric and Circular Functions

3 Standard Schedule Pacing Guide Da Section Suggested Assignment 6- Sinusoids: Amplitude, Period, and Ccles 9 RA, Q Q0, odd 6- General Sinusoidal Graphs 5, 5 odd, 6, (7), Graphs of Tangent, Cotangent, Secant, and Cosecant Functions RA, Q Q0,, 5, 6, 9, 4, (5, 6) Radian Measure of Angles RA, Q Q0,, 9,, 7,, 5, 9,, 7 5 odd Circular Functions Inverse Circular Relations: Given, Find RA, Q Q0, odd 8 Sinusoidal Functions as Mathematical RA, Q Q0,, Models 4, 5, 8,, (4, 5) 0 RA, Q-Q0, 9 odd, 0, 6-8 Rotar Motion 7 RA, Q Q0,, 6, 7, 0,, 4, 5, 9,, 5, 7,, 7, 9, 44, (45, 46, 48), 49 R0 R8, T T4 6-9 Chapter Review and Test Either C or C or Problem Set 7- Block Schedule Pacing Guide D a Section Suggested Assignment 6- General Sinusoidal Graphs RA, Q Q0, 5 odd, 6, Graphs of Tangent, Cotangent, Secant, and Cosecant Functions RA, Q Q0,, 5, 6, 9, Radian Measure of Angles RA, Q Q0,, 9,, 7, 6-4 Radian Measure of Angles 5, 9,, 7, 9, 4 5 odd 6-5 Circular Functions RA, Q Q0,, 6, 7, 0,, 4, 5, 9, 6-5 Circular Functions 5, 7,, 7, 44, Inverse Circular Relations: Given, Find RA, Q Q0, odd Sinusoidal Functions as Mathematical Models RA, Q Q0,,, 0, 4, Rotar Motion RA, Q Q0, 9 odd, 0,, Chapter Review and Test R0 R8, T T Chapter Review and Test 7- Introduction to the Pthagorean Propert 9 Chapter 6 Interleaf 8B

4 S e c t i o n 6 - Class Time da PLANNING Homework Assignment Problems 9 Teaching Resources Eploration 6-a: Sine and Cosine Graphs, Manuall TEACHING Important Terms and Concepts Ccle Amplitude Period Argument Phase displacement Sinusoidal ais Mathematical Overview GRAPHICALLY ALGEBRAICALLY NUMERICALLY You have learned that the graphs of cos and sin are sinusoids. You have also learned about horizontal and vertical dilations, and horizontal and vertical translations of functions. Now it is time to appl these four transformations to sinusoids so that ou can fit them to man real-world situations where the -values var periodicall. The graph is a sinusoid that is a cosine function transformed through vertical and horizontal translations and dilations. The independent variable here is rather than so that ou can fit sinusoids to situations that do not involve angles. Particular equation: 7 cos ( ) Section Notes Section 6- is an eplorator activit in which students investigate how the amplitude, period, phase displacement, vertical translation, and ccle of a sinusoidal graph are related to transformations of the parent sinusoid. You can assign Section 6- for homework after the Chapter 5 test or as a group activit to be completed in class. No classroom discussion is needed before students begin the activit. After the have completed Section 6-, have students work in groups on Eploration 6-a. Eploration Notes Eploration 6-a requires students to plot the sine and cosine graphs on graph paper. Problem 4 requires students to find values of inverse trigonometric functions. You ma wish to use this eploration as a quiz after Section 6- instead of using it with Section 6-. VERBALLY 8 Chapter 6: CAS Suggestions Consider having students who use TI-Nspire graphers define functions on a CAS and use function notation to reinforce the connection between the algebraic and graphical forms of transformations. The net figure shows the graphs in Problem 4 after defining f() 5 cos on a Calculator page. The circular functions are just like the trigonometric functions ecept that the independent variable is an arc of a unit circle instead of an angle. Angles in radians form the link between angles in degrees and numbers of units of arc length. Alternativel, students who use TI-Nspire graphers can graph 5 cos directl and manipulate the graph until the equation becomes the desired 5 cos. In this wa, students phsicall transform the parent function, strengthening the connection for kinesthic and visual learners. 8 Chapter 6: Applications of Trigonometric and Circular Functions

5 6- Sinusoids: Amplitude, Period, and Ccles Figure 6-a shows a dilated and translated sinusoid and some of its graphical features. In this section ou will learn how these features relate to transformations ou ve alread learned. Phase displacement (horizontal translation) Period Amplitude PROBLEM NOTES Problems 5, 7, and 9 can be solved using the approaches described in the CAS Suggestions.. Amplitude 5 5 One ccle Sinusoidal ais Objective Eplorator Problem Set 6- Section 6-: Figure 6-a Learn the meanings of amplitude, period, phase displacement, and ccle of a sinusoidal graph.. Sketch one ccle of the graph of the parent 5. Plot the graph of cos( 60 ). What sinusoid cos, starting at 0. What is transformation is caused b the 60? the amplitude of this graph? 6. The ( 60 ) in Problem 5 is called the. Plot the graph of the transformed cosine argument of the cosine. The phase displacement function 5 cos. What is the amplitude of is the value of that makes the argument equal this graph? What is the relationship between the zero. What is the phase displacement for this amplitude and the vertical dilation of a sinusoid? function? How is the phase displacement related to the horizontal translation? 7. Plot the graph of 6 cos. What transformation is caused b the 6? 8. Th e sinusoidal ais runs along the middle of the graph of a sinusoid. It is the dashed centerline in Figure 6-a. What transformation of the function cos does the location of the sinusoidal ais indicate?. What is the period of the transformed function in Problem? What is the period of the parent function cos? 4. Plot the graph of cos. What is the period of this transformed function graph? How is the related to the transformation? How could ou calculate the period using the? 9. What are the amplitude, period, phase displacement, and sinusoidal ais location of the graph of 6 5 cos ( 60 )? Check b plotting on our grapher.. Amplitude The absolute value of the vertical dilation of a sinusoid equals the amplitude for both functions The is the reciprocal of the horizontal dilation. The period equals 608 divided b. 5. Horizontal translation b Phase displacement The phase displacement equals the horizontal translation. 7. Vertical translation b 6 8. The vertical translation corresponds to the sinusoidal ais. 9. Amplitude 5 5; Period 5 08; Phase displacement 5 608; Sinusoidal ais Section 6-: Sinusoids: Amplitude, Period, and Ccles 8

6 Section 6- Class Time das PLANNING Homework Assignment Da : RA, Q Q0, Problems odd Da : Problems 5, 5 odd, 6, (7), 8 Teaching Resources Eploration 6-: Periodic Dail Temperatures Eploration 6-a: Transformed Sinusoid Graphs Eploration 6-b: Sinusoidal Equations from Graphs Blackline Masters Problems 5 8,, and 4 Technolog Resources Eploration 6-: Periodic Dail Temperatures TEACHING 6- Objective General Sinusoidal Graphs In Section 6-, ou encountered the terms period, amplitude, ccle, phase displacement, and sinusoidal ais. The are often used to describe horizontal and vertical translation and dilation of sinusoids. In this section ou ll make the connection between the new terms and these transformations so that ou will be able to fit an equation to an given sinusoid. This in turn will help ou use sinusoidal functions as mathematical models for real-world applications such as the variation of average dail temperature with the time of ear. Given an one of these sets of information about a sinusoid, find the other two: sinusoidal ais In this eploration ou will appl a transformed sinusoid to model the average dail high temperature at a particular location as a function of time. EXPLORATION 6-: Periodic Dail Temperatures This graph shows the average dail high temperatures in San Antonio, b month, for. Th e graph of cos completes a ccle each 60 (angle, not temperature!). Shown here is 4 months. The dots are connected b a smooth a transformed sinusoid that completes a ccle curve resembling a sinusoid. Month is Januar, each a horizontal dilation b a factor month is Februar, and so forth. of 0. Write the equation of this sinusoid and, Average high temperature, F plot it on our grapher. Does the result agree 00 with this figure? 50 Important Terms and Concepts Period Frequenc General sinusoidal equation Upper bound Lower bound Critical points Month, Eploration Notes Eploration 6- previews both the sinusoidal graph transformations in Section 6- and the real-world problems in Section 6-7. It is one of the most important eplorations in Chapter 6, for these reasons: Plotting points using pencil and paper helps students internalize the pattern of sinusoidal graphs. The use of real-world data demonstrates that sinusoidal graphs are relevant and useful, providing motivation for students to learn about them. 84 Chapter 6: An alternate version of this eploration can be found in the Instructor s Resource Book. Allow 0 5 minutes for all si questions. See page 89 for notes on additional eplorations.. 5 cos(0) Graph agrees with the given figure. (Be sure to use degree mode.). 5 cos(0( 78)) Graph agrees with given figure cos(0( 78)) Graph agrees with the given figure. continued 84 Chapter 6: Applications of Trigonometric and Circular Functions

7 EXPLORATION, continued. Th e greatest average high temperature occurs in Jul, month 7. Write an equation for a horizontal translation b 7 for the function in Problem and plot it on our grapher. Does the result agree with the graph shown here? Th e Jul average high temperature is 95 F. The lowest average high temperature is 6 F in Januar. So the vertical dilation of the (95 6) sinusoid is, or 7. Transform the equation in Problem so that it has a vertical dilation b a factor of 7 and plot it on our grapher. Does the result agree with this graph? Recall from Chapter 5 that the period of a sinusoid is the number of degrees per ccle. The reciprocal of the period, or the number of ccles per degree, is called the frequenc. It is convenient to use the frequenc when the period is ver short. For instance, the alternating electrical current in the United States has a frequenc of 60 ccles per second, meaning that the period is /60 second per ccle. You can see how the general sinusoidal equations allow for all four transformations. DEFINITION: General Sinusoidal Equation C A cos B( D) or C A sin B( D), where A is the amplitude (A is the vertical dilation, which can be positive or negative). B is the reciprocal of the horizontal dilation cos(0( 78)) Graph agrees with the given figure. 5. For March, month, For March, month 5, , the same. 6. Answers will var. C is the location of the sinusoidal ais (vertical translation). D is the phase displacement (horizontal translation) Section 6-: 4 4. Th e graph before Problem is a vertical translation of the graph in Problem b 95 7 (or 6 7), which equals 78. Write an equation for this translation of the function in Problem and plot it on our grapher. Does the result agree with the graph before Problem? 5. Predict the average high temperature in San Antonio for March, month. Do ou get the same answer for March, month 5? 6. What did ou learn as a result of doing this eploration that ou did not know before? 85 Section Notes Section 6- investigates equations of sinusoids, using the concepts of horizontal and vertical dilations and translations studied in the previous chapters. Students learn to write an equation for a sinusoidal function given an entire ccle of the graph, part of a ccle of the graph, or a verbal description of the graph. The also learn to write more than one equation for the same graph b using different phase displacements, negative dilations, and sine as well as cosine. It is recommended that ou spend two das on this section. On the first da, discuss the material up through Eample on page 88 and assign Eploration 6-a. Cover the remainder of the section on the second da, possibl using Eploration 6-b. Eample 4 on page 90 involves writing different equations for the same sinusoid. If our class is not ver strong, ou ma omit this eample, along with Problems and 4. However, encourage our best students to work through these items. When ou assign homework for this section, be sure to include problems that require students to write equations for sinusoids. Such problems prepare students for the real-world problems in Section 6-7. If a problem does not indicate which parent function to use, encourage students to write the equation using the cosine function. Because cosine starts a ccle at a high point, it is easier to get the correct phase displacement. It is also easier to find the general solution of an inverse cosine equation than of an inverse sine equation, so students will find the real-world problems in future sections easier to solve if the use the cosine function. Section 6-: General Sinusoidal Graphs 85

8 Section Notes (continued) Use the new vocabular introduced frequentl and correctl. Concavit, point of inflection, upper bound, and lower bound are all terms students will use in calculus. This section introduces the idea of frequenc, which is used to describe man real-world phenomena. It is most important that students understand the effects of the constants A, B, C, and D on the graphs of 5 C A cos B( D) and 5 C A sin B( D) Students usuall have the most difficult understanding the effect of the constant B. It is helpful to point out that B 5 60 period. Emphasize that the distance between a critical point and the net point of inflection is a quarter of a ccle, or period 4. This fact is useful for Eample on page 89 and Problems 5 8. From their earlier work with the parent cosine and sine functions, students should know that the cosine function starts a ccle at a high point, whereas the sine function starts a ccle on the sinusoidal ais going up. Stress that a sine function starts on its sinusoidal ais, not on the horizontal ais of the coordinate plane. Students often misunderstand this distinction because the sinusoidal ais of the parent sine graph coincides with the horizontal ais of the coordinate plane. Another error students make is marking a critical point or point of inflection on the -ais for ever sinusoidal graph. Emphasize that students must think about both the phase displace ment and the period when marking critical points and inflection points. 86 Chapter 6: The period can be calculated from the value of B. Because is the horizontal B dilation and because the parent cosine and sine functions have period 60, the period of a sinusoid equals (60 ). Dilations can be positive or negative, B so ou must use the absolute value smbol. PROPERTIES: Period and Frequenc of a Sinusoid For general equations C A cos B( D) or C A sin B( D) period B (60 ) and frequenc B period 60 Net ou ll use these properties and the general equation to graph sinusoids and find their equations. Concavit, Points of Inflection, and Bounds of Sinusoids In Section 4-4 ou learned about concavit and points of inflection. As ou can see from Figure 6-a, sinusoids have regions that are concave upward and other regions that are concave downward. The concavit changes at points of inflection. Concave side Conve side Points of inflection Concave up Figure 6-a Concave down As ou can see from Figure 6-b, the sinusoidal ais goes through the points of inflection. The lines through the high points and the low points are called the upper bound and the lower bound, respectivel. The high points and low points are called critical points because the have a critical influence on the size and location of the sinusoid. Note that it is a quarter-ccle between a critical point and the net point of inflection. Figure 6-b 86 Chapter 6: Applications of Trigonometric and Circular Functions

9 EXAMPLE SOLUTION Suppose that a sinusoid has period per ccle, amplitude 7 units, phase displacement 4 with respect to the parent cosine function, and a sinusoidal ais 5 units below the -ais. Without using our grapher, sketch this sinusoid and then find an equation for it. Verif with our grapher that our equation and the sinusoid ou sketched agree with each other. First draw the sinusoidal ais at 5, as in Figure 6-c. (The long-and-short dashed line is used b draftspersons for centerlines.) Use the amplitude, 7, to draw the upper and lower bounds 7 units above and 7 units below the sinusoidal ais. 5 Upper bound Lower bound Figure 6-c Net find some critical points on the graph (Figure 6-d). Start at 4, because that is the phase displacement, and mark a high point on the upper bound. (The cosine function starts a ccle at a high point because cos 0.) Then use the period,, to plot the ends of the net two ccles ( ) 0 Mark some low critical points halfwa between consecutive high points Figure 6-d 0 Now mark the points of inflection (Figure 6-e). The lie on the sinusoidal ais, halfwa between consecutive high and low points Figure 6-e Domain 0 θ Section 6-: Range Eample 5 all real numbers of degrees Eample 5 all real numbers of degrees 8 56 Eamples and 4 5 all real numbers of degrees 87 To prepare students for Problems 5 8, ou might ask them to evaluate the function in Eample on page 88 for Show students how to find numericall b using the table feature on their graphers and algebraicall b using function notation. f () cos 8( ) f (595 ) cos 8(595 ) f (595 ) One eplanation for wh both sine and cosine work for the equation in Eample is the cofunction relationship between the two functions; that is, sin 5 cos(90 ) for all acute angle values of. This relationship can now be etended to all angle values for. In Eample 4 on page 90, students use graphers to confirm that all four equations give the same graph. If their graphers allow it, have students use different line stles for each graph. Some graphing calculators can be set to displa a circle that follows the path created b the graph. Using this feature allows students to see that each subsequent graph retraces the preceding graphs. Ask students to identif the domain and range of the functions in Eamples 4 as shown in the chart at the bottom of this page. Emphasize that the domain is all real numbers of degrees. Specifing degrees is important because real numbers without degrees are assumed to be radians. Section 6-: General Sinusoidal Graphs 87

10 Section Notes (continued) As an interesting assignment, ou might ask each student to find a photograph or a computer image of a sinusoidal curve and write a reasonable equation for the curve. Encourage students to find out what might be reasonable units for their photos. If ou show a few eamples in class, students will see that such curves are common and will be able to find eamples on their own. Create a bulletin board of the photos and equations students find. You ma want to give this assignment the da ou begin Chapter 6, but schedule its due date for after ou finish Section 6-7. As students stud each new section, the will get better ideas for possible pictures of sinusoids. This will also bring up the need for a nondegree angle measure. Eamples of naturall occurring sinusoids abound. Aerial photos of rivers and the crests of desert sand dunes form sinusoidal curves. Each monthl issue of the magazine Astronom contains a colorful page spread of the sinusoidal path of the moon. In addition, each month the sinusoidal paths of four of Jupiter s moons Io, Callisto, Europa, and Ganmede are shown. The book Mathematics: An Introduction to Its Spirit and Use has photographs of musical notes recorded b an oscilloscope. The fundamental harmonic has one ccle; the second harmonic has two ccles; the third harmonic has three ccles; and the fourth harmonic has four ccles. The December 996 issue of National Geographic shows places on Earth photographed during 0 ears of space flights and also shows the path of a tpical spacecraft. Distinctive sinusoidal paths are visible in these photographs. EXAMPLE 88 Chapter 6: Finall, sketch the graph in Figure 6-f b connecting the critical points and points of inflection with a smooth curve. Be sure that the graph is rounded at the critical points and that it changes concavit at the points of inflection Figure 6-f Because the period of this sinusoid is and the period of the parent cosine function is 60, the horizontal dilation is dilation 60 0 The coefficient B in the sinusoidal equation is the reciprocal of 0, namel, 0. The horizontal translation is 4. Thus a particular equation is 5 7 cos 0( 4 ) Plotting the graph on our grapher confirms that this equation produces the correct graph (Figure 6-g). Figure 6-g For the sinusoid in Figure 6-h, give the period, frequenc, amplitude, phase displacement, and sinusoidal ais location. Write a particular equation for the sinusoid. Check our equation b plotting it on our grapher Differentiating Instruction Pass out the list of Chapter 6 vocabular, available at keonline, for ELL students to look up and translate in their bilingual dictionaries. Figure 6-h Use several eamples of argument, bound, and frequenc and get student responses to make sure the understand these important mathematical meanings. Bilingual dictionaries ma not include their mathematical definitions. Create graphs with the u- and v-aes scaled from [4, 4]. On two of the sets of aes have graphs of sine and cosine. The paper graphs will enable students 88 Chapter 6: Applications of Trigonometric and Circular Functions

11 SOLUTION EXAMPLE SOLUTION As ou will see later, ou can use either the sine or the cosine as the pre-image function. Here, use the cosine function, because its first ccle starts at a high point and two high points are known. and ends at, so the period is, or 0. ccle per degree. _ (8 56), or 9. sinusoidal ais. A is. (You could also use or 7.) 0 60, so B 60, or 8 (the reciprocal of the 0 horizontal dilation). So a particular equation is 9 47 cos 8( ) Plotting the corresponding graph on our grapher confirms that the equation is correct. You can find an equation of a sinusoid when onl part of a ccle is given. The net eample shows ou how to do this. Figure 6-i shows a quarter-ccle of a sinusoid. Write a particular equation and check it b plotting it on our grapher. 8 7 Figure 6-i 4 Imagine the entire ccle from the part of the graph that is shown. You can tell that a low point is at 4 because the graph appears to level out there. So the lower bound is at. The point at 7 must be an inflection point on the sinusoidal ais at 8 because the graph is a quarter-ccle. So the amplitude is 8, or 5. Sketch the lower bound, the sinusoidal ais, and the upper bound. Net locate a high point. to trace with their fingers to reinforce their understanding and to enable ou to verif that the understanding does (or doesn t) eist. These graphs can also help reinforce the meanings of sinusoid, periodic, and displacement, and clarif translation of one period, pre-image and image graphs, and negative angles. Have students use their paper graphs while ou go over the eamples. 0 Section 6-: 89 Have students write the general sinusoidal equation definition in their journals both formall and in their own words. Make sure the distinction on page 90 between a particular equation and the particular equation is clear. Consider using the Reading Analsis as an in-class discussion. Alternativel, allow ELL students to do it in pairs. Refer to the Problem Notes for Problems 5 8 and Problems 9 8. These problems will help prepare students for Section 6-7, which will present a challenge for ELL students because of the comple language. Problem 8 will challenge ELL students to develop their English skills. Additional Eploration Notes Eploration 6-a gives students practice in graphing particular cosine and sine equations; identifing period, amplitude, and other characteristics; writing an equation from a given graph; checking their work using a grapher; and finding a -value given an angle value. Assign this eploration as a group activit after ou have presented and discussed Eamples and. This gives students supervised practice with the ideas presented in the eamples before the leave class. Students ma need help with Problem 5, which asks them to find when 5 5. Allow 0 0 minutes for the eploration. Eploration 6-b can be assigned as a quiz or as etra homework practice. It can also be assigned later in the ear as a review of sinusoids. Problem of the eploration asks students to sketch two ccles of a given sinusoid. Problems and ask them to write two equations one using sine and one using cosine for the same sinusoid and then to check their equations using their graphers. Problems 5 and 6 give students practice writing equations based on a half-ccle or quarter-ccle of a sinusoid. Allow 0 minutes for the entire eploration or 8 0 minutes for just Problems 5 and 6. Section 6-: General Sinusoidal Graphs 89

12 Technolog Notes Eploration 6- asks students to transform a cosine graph to fit data on the average dail temperature for a cit. This can be done easil with sliders in Fathom. CAS Suggestions Students using a CAS can determine the horizontal dilation b remembering that the period of cosine was multiplied b the horizontal dilation to determine the period of the new function. Note: The algebra in this approach can be done b hand. However, the point of this problem is not the manipulation, but the meaningful determination of the coefficients; a CAS is entirel appropriate for this work. Once an equation is determined, it can be tested to see if the period is properl represented algebraicall. Define the equation. Then test that equation for the given period. While this does not absolutel confirm that the period has been correctl computed, it does confirm whether an equation has a period that is an integer multiple of the correct period. Students can use Boolean logic to confirm the equivalence of the equations in Eample 4 on page 90. EXAMPLE 4 Each quarter-ccle covers (4 7 ), or 7, so the critical points and points of inflection are spaced 7 apart. Thus a high point is at 7 7, or 0. Sketch at least one complete ccle of the graph (Figure 6-j) Figure 6-j 4 The period is 4(7 ), or 8, because a quarter of the period is 7. The horizontal dilation is 8 60, or The coefficient B in the sinusoidal equation is the reciprocal of this horizontal dilation. If ou use the techniques of Eample, a particular equation is 8 5 cos 90 ( 0 ) 7 Plotting the graph on our grapher shows that the equation is correct. Note that in all the eamples so far a particular equation is used, not the. There are man equivalent forms of the equation, depending on which ccle ou pick for the first ccle and whether ou use the parent sine or parent cosine function. The net eample shows some possibilities. For the sinusoid in Figure 6-k, write a particular equation using a. Cosine, with a phase displacement other than 0 b. Sine c. Cosine, with a negative vertical dilation factor d. Sine, with a negative vertical dilation factor Confirm on our grapher that all four equations give the same graph Figure 6-k Chapter 6: 90 Chapter 6: Applications of Trigonometric and Circular Functions

13 SOLUTION a. Notice that the sinusoid is the same one as in Eample. To find a different phase displacement, look for another high point. A convenient one is at 8. All the other constants remain the same. So another particular equation is 8 5 cos 90 ( 8 ) 7 b. The graph of the parent sine function starts at a point of inflection on the sinusoidal ais while going up. Two possible starting points appear in Figure 6-k, one at and another at. 8 5 sin 90 7 ( ) or 8 5 sin 90 ( ) 7 c. Changing the vertical dilation factor from 5 to 5 causes the sinusoid to be reflected across the sinusoidal ais. If ou use 5, the first ccle starts as a low point instead of a high point. The most convenient low point in this case is at cos 90 ( 4 ) 7 d. With a negative dilation factor, the sine function starts a ccle at a point of inflection while going down. One such point is shown in Figure 6-k at sin 90 ( 7 ) 7 Plotting these four equations on our grapher reveals onl one image. The graphs are superimposed on one another. Q. ccles Q. 8 Q. 08 Q4. PROBLEM NOTES Q5. 08 for 5 cos Q6. Q Q Problem Set 6- Reading Analsis From what ou have read in this section, what do ou consider to be the main idea? How are the period, frequenc, and ccle related to one another in connection with sinusoids? What is the difference between the wa appears on the graph of a sinusoid and the wa it appears in a uv-coordinate sstem, as in Chapter 5? How can there be more than one particular equation for a given sinusoid? 5min Quick Review Problems Q Q5 refer to Figure 6-l. Q. How man ccles are there between 0 and 80? Q. What is the amplitude? Figure 6-l Q. What is the period? Q4. What is the vertical translation? Q5. What is the horizontal translation (for cosine)? Q6. Find the eact value (no decimals) of sin 60. Q7. Find the approimate value of sec 7. Q8. Find the approimate value of co t 4.. Section 6-: 9 Section 6-: General Sinusoidal Graphs 9

14 Problem Notes (continued) Q9. tan Q Q9. Find the measure of the larger acute angle of a right triangle with legs of lengths ft and 9 ft. Q0. Epand: ( 5) 6. 0 and 45 8 Problems 4 should all be assigned so students get sufficient practice in graphing transformed sinusoids.. Amplitude 5 4; Period 5 08; Phase displacement 5 08; Sinusoidal ais and Amplitude 5 5; Period ; Phase displacement 5 408; Sinusoidal ais Amplitude 5 0; Period 5 708; Phase displacement 5 08; Sinusoidal ais For Problems 4, find the amplitude, period, phase displacement, and sinusoidal ais location. Without using our grapher, sketch the graph b locating critical points. Then check our graph using our grapher cos ( 0 ). 5 cos _ 4 ( 40 ). 0 0 sin _ ( 0 ) sin 5( 6 ) For Problems 5 8, a. Find a particular equation for the sinusoid using cosine or sine, whichever seems easier. b. Give the amplitude, period, frequenc, phase displacement, and sinusoidal ais location. c. Use the equation from part a to calculate for the given values of. Show that the result agrees with the given graph for the first value and and For Problems 9 4, find a particular equation for the sinusoid that is graphed Amplitude 5 0; Period 5 78; Phase displacement 5 68; Sinusoidal ais The Solve command could be used to determine the coefficient of in Problems Chapter 6: Problems 5 8 are important not onl because students must create equations for given graphs but also because the must fi n d -values for given angle values. This is an ecellent preview of Section 6-7, in which students use sinusoidal equations to predict results in real-world problems. Be sure to discuss finding the -values during the class discussion of the homework. A blackline master for these problems is available in the Instructor s Resource Book a cos ( 08) 5b. Amplitude 5 6; Period 5 808; Frequenc 5 80 ccle/deg; Phase displacement 5 08; Sinusoidal ais 5 9 5c at 5 608; at Chapter 6: Applications of Trigonometric and Circular Functions

15 In Problems 5 and 6, a half-ccle of a sinusoid is shown. Find a particular equation for the sinusoid r a cos 9( 4) 6b. Amplitude 5 0; Period 5 40; Frequenc 5 40 ccle/deg; Phase displacement 5 4; Sinusoidal ais 5 8 6c at 5 0; at a. 5 5 cos ( 0) 7 In Problems 7 and 8, a quarter-ccle of a sinusoid is shown. Find a particular equation for the sinusoid If the sinusoid in Problem 7 is etended to 00, what is the value of? If the sinusoid is etended to 5678, is the point on the graph above or below the sinusoidal ais? How far? 0. If the sinusoid in Problem 8 is etended to the left to.5, what is the value of? If the sinusoid is etended to 8, is the point on the graph above or below the sinusoidal ais? How far? For Problems and, sketch the sinusoid described and write a particular equation for it. Check the equation on our grapher to make sure it produces the graph ou sketched.. The period equals 7, the amplitude is units, the phase displacement (for cos ) equals 6, and the sinusoidal ais is at 4 units.. The frequenc is 0 ccle per degree, the amplitude equals units, the phase displacement (for cos ) equals, and the sinusoidal ais is at 5 units. Section 6-: 9 7b. Amplitude 5 5; Period 5 0; Frequenc 5 0 ccle/deg; Phase displacement 5 0; Sinusoidal ais 5 7c. 5 8 at 5 70; at a sin 8( ) 8b. Amplitude 5 0; Period 5 0; Frequenc 5 0 ccle/deg; Phase displacement 5 ; Sinusoidal ais 5 0 8c. 5 0 at 5 8; at Problems 9 8 provide practice in writing equations from graphs that show complete ccles or parts of ccles. This is another critical skill required to solve the real-world problems in Section 6-7. A blackline master for these problems is available in the Instructor s Resource Book sin 0( 6) cos 60( 0.). 5.7 cos( 0) cos 45( ) Problems and 4 use different variables to name the horizontal and vertical aes. When ou discuss homework, make sure students have used the correct variables and not just the usual variables and.. r 5 7 cos sin sin sin sin 9 ( 60) sin 45 Problems 9 and 0 require students to find -values corresponding to given angle values at 5 00; , below the sinusoidal ais at at 5.5; 5 60, on the sinusoidal ais, at 5 8 Problems and require students to write equations for sinusoids based on verbal descriptions. See page 00 for answers to Problems and. Section 6-: General Sinusoidal Graphs 9

16 Problem Notes (continued) Problems and 4 ask students to write four possible equations for each graph b using both positive and negative dilations of both sine and cosine equations. These problems will challenge our best students. A blackline master for these problems is available in the Instructor s Resource Book. Problems and 4 call for the development of multiple equations for the given figures and graphical confirmation of their equivalence. Students can also use a CAS to provide Boolean logic to confirm the algebraic form of the derived equations. Problem 5 gives students practice determining frequenc. 5a. 60 ccles/deg. Thinking in terms of complete ccles (60 of them) gives a clearer mental picture than thinking in terms of fractions. 5b. Period 5.8; Frequenc 5 5_ ccle/deg. The frequenc 6 is 00 divided b 608. Problem 6 rein forces the ideas of inflection point, concave up, and concave down. These ideas are ver important in the stud of calculus. Be sure to assign this problem. Problem 7 provides practice in transforming a given equation to another form b using algebra. The final form of the equation helps students see that horizontal and vertical transformations follow the same rules. Problem 8 is a journal entr that gives students practice with the vocabular in Section 6-. You might tell the class that ou will randoml choose five students to read their journal entries to the class. 8. Journal entries will var. See pages for answers to Problems, 4, 6, and 7, and CAS Problems 4. For Problems and 4, write four different particular equations for the given sinusoid, using a. Cosine as the parent function with positive vertical dilation b. Cosine as the parent function with negative vertical dilation c. Sine as the parent function with positive vertical dilation d. Sine as the parent function with negative vertical dilation Plot all four equations on the same screen on our grapher to confirm that the graphs are the same Frequenc Problem: The unit for the period of a sinusoid is degrees per ccle. The unit for the frequenc is ccles per degree. a. Suppose a sinusoid has period 60 degree/ccle. What would the frequenc be? Wh might people prefer to speak of the frequenc of such a sinusoid rather than the period? b. For cos 00, what is the period? What is the frequenc? How can ou calculate the frequenc quickl, using the 00? 6. Inflection Point Problem: Sketch the graph of a function that has high and low critical points. On the sketch, show a. A point of inflection b. A region where the graph is concave up c. A region where the graph is concave down 94 Chapter 6: Additional CAS Problems. Describe the graphical transformations required to change 5 cos into 5 cos(908 ). Eplain how this confirms the relationship sin 5 cos(908 ) for all angle values of.. Enter cos(908 ) into our CAS. What is the output and wh does it confirm that the relationship 7. Horizontal vs. Vertical Transformations Problem: In the function 4 cos ( 5 ) the and the 4 are the vertical transformations, but the and the 5 are the reciprocal and opposite of the horizontal transformations. a. Show that ou can transform the given equation to cos 5 4 / b. Eamine the equation in part a for the transformations that are applied to the - and -variables. What is the form of these transformations? c. Wh is the original form of the equation more useful than the form in part a? 8. Journal Problem: Update our journal with things ou have learned about sinusoids. In particular, eplain how the amplitude, period, phase displacement, frequenc, and sinusoidal ais location are related to the four constants in the general sinusoidal equation. What is meant b critical points, concavit, and points of inflection? sin 5 cos(908 ) holds for all values of?. If ou entered Solve( = f(), ) = 0 into our CAS where f () was an defined sinusoidal function, how man answers would ou get? Eplain. 4. If ou entered Solve( = f(), ) = 0 into our CAS where f () was an defined sinusoidal function, how man answers would ou get? Eplain. 94 Chapter 6: Applications of Trigonometric and Circular Functions

17 6- Objective Graphs of Tangent, Cotangent, Secant, and Cosecant Functions If ou enter tan 90 into our calculator, ou will get an error message because tangent is defined as a quotient. On the unit circle, a point on the terminal side of a 90 angle has horizontal coordinate zero and vertical coordinate. Division of an number b zero is undefined, which ou ll see leads to vertical asmptotes at angle measures for which division b zero would occur. In this section ou ll also see that the graphs of the tangent, cotangent, secant, and cosecant functions are discontinuous where the function value would involve division b zero. You can plot cotangent, secant, and cosecant b using the fact that the are reciprocals of tangent, cosine, and sine, respectivel. cot tan sec cos csc sin Figure 6-a shows the graphs of tan and cot, and Figure 6-b shows the graphs of sec and csc, all as the might appear on our grapher. If ou use a window that includes multiples of 90 as grid points, ou ll see that the graphs are discontinuous. Notice that the graphs go off to infinit (positive or negative) at odd or even multiples of 90, eactl those places where the functions are undefined.a tan cot Figure 6-a sec csc Figure 6-b To see wh the graphs have these shapes, it is helpful to look at transformations performed on the parent cosine and sine graphs. 95 Section 6- Class Time da PLANNING Homework Assignment RA, Q 0, Problems, 5, 6, 9, 4, (5, 6) Teaching Resources Eploration 6-a: Tangent and Secant Graphs Eploration 6-b: Transformed Tangent and Secant Graphs Supplementar Problems Test 5, Sections 6- to 6-, Forms A and B Technolog Resources Variation of Tangent and Secant Presentation Sketch: Trig Tracers Present.gsp Eploration 6-a: Tangent and Secant Graphs TEACHING Important Terms and Concepts Unit circle Vertical asmptote Discontinuous Quotient properties for tangent and cotangent Section Notes Section 6- introduces the graphs of the tangent, cotangent, secant, and cosecant functions. If ou prefer, ou can wait until after Section 6-7 to cover this section because the topics in Section 6-4 through Section 6-7 do not depend on the graphs of these four functions. If ou do reorder the sections, do not assign an Review or Quick Review problems that involve the graphs of the tangent, cotangent, secant, or cosecant functions. Section 6-: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions 95

18 Section Notes (continued) Another important outcome of this section is introducing the quotient properties to students before the encounter them in Chapter 7. The graphs of the tangent, cotangent, secant, and cosecant functions are shown near the beginning of the section. When working with functions that have vertical asmptotes, it can be helpful on some graphers to use a friendl window where the horizontal range is a decimal multiple of 94. Also, the -min should be a multiple of 90. For eample, the graphs of 5 tan and 5 sec could have a window Note that ma min (70) , so each piel move represents a change of 0 in. This can be helpful in tracing the graph of the function. The Section Notes for Section - contain more information on friendl graphing windows. The tet assumes that students need to use the reciprocal properties to graph and calculate cosecant, secant, and cotangent on their graphers, but note that some graphers will evaluate and graph these functions directl. The graphs in Figures 6-a and 6-b can be replicated on a grapher with viewing ranges 70 60, 6 6 for 5 tan and 5 sec, and 80 70, 6 6 for 5 cot and 5 csc. All four graphs have -scale 90 and -scale. Before giving students the window dimensions, encourage them to eplore the domain and range of functions in order to find appropriate viewing windows. Make sure their graphers are set in degree mode for this section. Notice that these are not friendl windows, which were mentioned earlier. As a result, students with older graphers (such as TI-8) will see that the top of one section of a graph is connected to the bottom of the net section, where there should be an asmptote. EXAMPLE 96 Chapter 6: SOLUTION EXAMPLE This can be partiall remedied b pressing the MODE button and selecting DOT rather than CONNECTED, or it can be full remedied b using a friendl graphing window, as discussed earlier. In the eamples, eplorations, and problems, students learn how to sketch these graphs b relating them to the sine and cosine graphs. For eample, because sec 5, the graph of 5 sec is cos undefined where cos 5 0, has value 96 Chapter 6: Applications of Trigonometric and Circular Functions Sketch the graph of the parent sine function, sin. Use the fact that csc to sketch the graph of the cosecant function. Show how the sin asmptotes of the cosecant function are related to the graph of the sine function. Sketch the sine graph as in Figure 6-c. Where the value of the sine function is zero, the cosecant function will be undefined because of division b zero. Draw vertical asmptotes at these values of. Figure 6-c 60 Where the sine function equals or, so does the cosecant function, because the reciprocal of is and the reciprocal of is. Mark these points as in Figure 6-d. As the sine gets smaller, the cosecant gets larger, and vice versa. For instance, the reciprocal of 0. is 5, and the reciprocal of 0. is 0. Sketch the graph consistent with these facts, as in Figure 6-d. To understand wh the graphs of the tangent and cotangent functions have the shapes in Figure 6-a, it is helpful to eamine how these functions are related to the sine and cosine functions. If is the standard position angle of a ra from the origin to point (u, v) in a uv-coordinate sstem, and r is the distance from the origin to (u, v), then b definition, tan u v Dividing the numerator and the denominator b r gives tan v/r u/r B the definitions of sine and cosine, the numerator equals sin and the denominator equals cos. As a result, these quotient properties are true. PROPERTIES: Quotient Properties for Tangent and Cotangent tan sin cos and cot cos sin Figure 6-d The quotient properties allow ou to construct the tangent and cotangent graphs from the sine and cosine graphs. On paper, sketch the graphs of sin and cos. Use the quotient propert to sketch the graph of cot. Show the asmptotes and the points where the graph crosses the -ais. 60 where cos 5, and has value where cos 5. When ou present the secant graph, sketch a graph of the cosine function and show that the secant graph bounces off the high and low points of the cosine graph because secant is the reciprocal of cosine. Similarl, because cosecant is the reciprocal of sine, the cosecant graph bounces off the high and low points of the sine graph.

19 SOLUTION 60 Draw the graphs of the sine and cosine functions (dashed and solid, respectivel) as in Figure 6-e. Because cot cos, show the asmptotes where sin 0, and sin show the -intercepts where cos 0. At 45, and wherever else the graphs of the sine and the cosine functions intersect each other, cos will equal. Wherever sine and cosine are opposites sin of each other, cos will equal. Mark these points as in Figure 6-f. Then sketch sin the cotangent graph through the marked points, consistent with the asmptotes. The final graph is shown in Figure 6-g Figure 6-f Figure 6-g Reading Analsis Q8. What kind of function is 5.? From what ou have read in this section, what do ou consider to be the main idea? What feature do the graphs of the tangent, cotangent, secant, and cosecant functions have that sinusoids do not have, and wh do the have this feature? What algebraic properties allow ou to sketch the graph of the tangent or cotangent function from two sinusoids? 5min Figure 6-e Problem Set 6- Quick Review Problems QQ7 refer to the equation 4 cos 5( 6 ). Q. The graph of the equation is called a?. Q. The amplitude is?. Q. The period is?. Q4. The phase displacement with respect to cos is?. Q5. The frequenc is?. Q6. The sinusoidal ais is at?. Q7. The lower bound is at?. 60 Q9. What kind of function is 5? Q0. The If part of the statement of a theorem is called the A. Conclusion B. Hpothesis C. Converse D. Inverse E. Contrapositive. Secant Function Problem a. Sketch two ccles of the parent cosine function, cos. Use the fact that sec to sketch the graph of sec. cos b. How can ou locate the asmptotes in the secant graph b looking at the cosine graph? How does our graph compare with the secant graph in Figure 6-b? c. Does the secant function have critical points? If so, find some of them. If not, eplain wh not. d. Does the secant function have points of inflection? If so, find some of them. If not, eplain wh not. Function Period Domain Range 5 tan 80 5 all real numbers of degrees ecept n, where n is an integer 5 cot 80 5 all real numbers of degrees ecept 5 80 n, where n is an integer 5 sec 60 5 all real numbers of degrees ecept n, where n is an integer 5 csc 60 5 all real numbers of degrees ecept 5 80 n, where n is an integer 97 5 all real numbers 5 all real numbers or or Differentiating Instruction Discuss Problem 7 so students understand wh the period of the tangent and cotangent functions is 80 rather than 60. Students ma need to be reminded of this fact throughout the ear. Net, discuss Problem 9, which asks students to find the domain of 5 sec. Have students list several angle values at which there are asmptotes. Eventuall someone should point out that the asmptotes appear 80 apart and will attempt to state the domain. Once the domain is given, have students eamine the graphs of the tangent, cotangent, and cosecant functions. Point out the general pattern where is undefined in the domain: 5 (location of st asmptote) (distance between asmptotes) n, where n is an integer. Summarize the section using the chart at the bottom of this page. Eploration Notes Eploration 6-a requires students to graph tangent and secant functions and to write equations for graphs of tangents and cosecants. You ma want to assign this eploration as additional homework practice if ou do not have time to use it in class. Eploration 6-b requires students to graph tangent and cosecant equations and to write tangent and secant equations for graphs. You ma want to assign this eploration as additional homework practice if ou do not have time to use it in class. See page 00 for answers to Problems Q Q0 and Problem. Section 6-: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions 97

20 Technolog Notes Problem 6 asks students to eplore the behaviors of the si trigonometric functions as the angle of measurement varies. The ma create their own sketches using Sketchpad, or the ma use the Variation of Tangent and Secant eploration at Presentation Sketch: Trig Tracers Present.gsp at is a presentation sketch that traces out the graphs of the si trigonometric functions as a point is animated on a unit circle. Eploration 6-a: Tangent and Secant Graphs has students discover what the graphs of tangent and secant look like and how the relate to the graphs of sine and cosine. The better resolution available in Fathom ma be useful for the parts where the are asked to use a grapher. Sketchpad could also be emploed here. PROBLEM NOTES Supplementar problems for this section are available at keonline. Problems 4 reinforce the work done in Eploration 6-a.. Tangent Function Problem a. Sketch two ccles of the parent function cos and two ccles of the parent function sin on the same aes. b. Eplain how ou can use the graphs in part a to locate the -intercepts and the vertical asmptotes of the graph of tan. c. Mark the asmptotes, intercepts, and other significant points on our sketch in part a. Then sketch the graph of tan. How does the result compare with the tangent graph in Figure 6-a? d. Does the tangent function have critical points? If so, find some of them. If not, eplain wh not. e. Does the tangent function have points of inflection? If so, find some of them. If not, eplain wh not.. Quotient Propert for Tangent Problem: Plot these three graphs on the same screen on our grapher. Eplain how the result confirms the quotient propert for tangent. f () sin f () cos f () f ()/ f () 4. Quotient Propert for Cotangent Problem: On the same screen on our grapher, plot these three graphs. Eplain how the result confirms the quotient propert for cotangent. f () sin f () cos f () f ()/f () 5. Without referring to Figure 6-a, quickl sketch the graphs of tan and cot. 6. Without referring to Figure 6-b, quickl sketch the graphs of sec and csc. 7. Eplain wh the period of the functions tan and cot is onl 80, instead of 60 like the periods of the other four trigonometric functions. 8. Eplain wh it is meaningless to talk about the amplitude of the tangent, cotangent, secant, and cosecant functions. 9. What is the domain of the function sec? What is its range? 0. What is the domain of the function tan? What is its range? For Problems 4, what are the dilation and translation caused b the constants in the equation? Plot the graph on our grapher and show that these transformations are correct.. 5 tan ( 5 ). cot ( 0 ). 4 6 sec _ ( 50 ) 4. csc 4( 0 ) 5. Rotating Lighthouse Beacon Problem: Figure 6-h shows a lighthouse located 500 m from the shore. Light ra Lighthouse beacon Spot of light 500 m D Other light ra Figure 6-h Shore In Problem b, the -intercepts and vertical asmptotes can be found on a CAS b solving equations involving the corresponding portions of the tangent fraction. Notice that the vertical asmptote locations are written in an uncommon, but correct, form. Recognizing equivalent forms is a critical mathematical skill, especiall when using a CAS. 98 Chapter 6: a., c b. Asmptotes occur where cos 5 0 at n; -intercepts occur where sin 5 0 at 5 808n. 98 Chapter 6: Applications of Trigonometric and Circular Functions

21 A rotating light on top of the lighthouse sends out ras of light in opposite directions. As the beacon rotates, the ra at angle makes a spot of light that moves along the shore. As increases beond 90, the other ra makes the spot of light. Let D be the displacement of the spot of light from the point on the shore closest to the beacon, with the displacement positive to the right and negative to the left as ou face the beacon from the shore. a. Plot the graph of D as a function of. Use a window with 0 60 and 000 D 000. Sketch the result. b. Where does the spot of light hit the shore when 55? When 9? c. What is the first positive value of for which D equals 000? For which D equals 000? d. Eplain the phsical significance of the asmptote at Variation of Tangent and Secant Problem: Figure 6-i shows the unit circle in a uv-coordinate sstem and a ra from the origin, O, at an angle,, in standard position. The ra intersects the circle at point P. A line is drawn tangent to the circle at point P, intersecting the u-ais at point A and the v-ais at point B. A vertical segment from point P intersects the u-ais at point C, and a horizontal segment from point P intersects the v-ais at point D. D 0.5 O v B P Movable point P C A u Figure 6-i Section 6-: c. See answer to part a. d. No, the graph is alwas increasing, where it is defined. e. Yes, at 5 80n Problems 5 0 provide the practice necessar for students to learn the graphs. 5. See Figure 6-a. 6. See Figure 6-b. a. Use the properties of similar triangles to eplain wh these segment lengths are equal to the si corresponding function values: PA tan PB cot PC sin PD cos OA sec OB csc b. The angle between the ra and the v-ais is the complement of angle ; that is, its measure is 90. Show that in each case the cofunction of is equal to the function of the complement of. c. Construct Figure 6-i using dnamic geometr software such as The Geometer s Sketchpad, or use the Variation of Tangent and Secant eploration at Observe what happens to the si function values as changes. Describe how the sine and cosine var as is made larger or smaller. Based on the figure, eplain wh the tangent and secant become infinite as approaches 90 and wh the cotangent and cosecant become infinite as approaches For all, where n is an integer sin( 80n) 5 sin and cos( 80n) 5 cos. Therefore, for all : sin ( 80n) tan( 80n) 5 cos ( 80n) 5 sin cos 5 sin 5 tan. So the period of the tangent cos function is 80. Furthermore, for all : cot( 80n) 5 cos ( 80n) sin ( 80n) 5 cos sin 5 cos 5 cot. sin 8. These functions have no maimum or minimum. 9. The domain of sec is where cos is defined, i.e., all 90 80n. The range is, i.e., (, ] [, ). 0. The domain of tan is where sin cos is defined, i.e., all 90 80n. The range is all real numbers. Problems 4 require students to consider transformations of the parent tangent, cotangent, secant, and cosecant graphs. The instructions ask students to make the graphs on their graphers, but the equations in Problems and 4 can also be sketched b hand using the technique demonstrated in Eample on page 96. For eample, to sketch the equation in Problem, replace the sec in the equation with cos and sketch the graph of the resulting equation. Then draw asmptotes where the sinusoid crosses its sinusoidal ais, draw the upward branch of the secant function where the sinusoid reaches its maimum value, and draw the downward branch of the secant function where the sinusoid reaches its minimum value. Problem 5 ma cause students difficult. You ma want to solve this as a wholeclass activit rather than assign it as homework. Because future problems do not rel on this problem, it can be omitted. Point out to students that ou are distinguishing between the two light beams and keeping track of which one is pointing in the positive direction and which one is pointing in the negative direction as it rotates. The terminal side of u is the positive direction. Thus, as the beacon rotates past 90, L becomes negative because it is the negativel oriented beam of light that hits the shore. See pages for answers to Problems, 4, and 6. Section 6-: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions 99

22 Section 6-4 Class Time da PLANNING Homework Assignment RA, Q Q0, Problems, 9,, 7,, 5, 9,, 7, 9, 4, 4, 45, 47, 49, 5, Radian Measure of Angles With our calculator in degree mode, press sin 60. You get sin Now change to radian mode and press sin. You get the same answer! sin In this section ou will learn what radians are and how to convert angle measures between radians and degrees. The radian measure of angles allows ou to epand on the concept of trigonometric functions, as ou ll see in the net section. Through this epansion of trigonometric functions, ou can model real-world phenomena in which independent variables represent distance, time, or an other quantit, not just an angle measure in degrees. Teaching Resources Eploration 6-4: Introduction to Radians Eploration 6-4a: Radian Measure of Angles Blackline Masters Problems, 49, and 50 Technolog Resources Presentation Sketch: Radians Present.gsp CAS Activit 6-4a: Inverse Trigonometric Functions TEACHING Important Terms and Concepts Radian Unit circle Subtends Unitless number Dimensional analsis Wrapping function Arc length Eploration Notes Eploration 6-4 is an ecellent activit for groups of two or three students. It gives each student an opportunit to measure radians on a circle. If ou have time for onl one eploration, do Eploration 6-4. Show students how to hold the inde card on its edge and bend it to follow the curve of the circle. Allow 0 5 minutes for this activit. See page 04 for notes on additional eplorations. 00 Chapter 6: Objective In this eploration ou will eplore the radian as a unit of angular measure, and ou ll see how this unit arises naturall from the radius of a circle. EXPLORATION 6-4: Introduction to Radians This graph shows a circle of radius r units in a uv-coordinate sstem. The radius has been marked off in units of tenths of a radius. An arc of the circle with a curved length eactl one radius unit long has been marked off starting at the positive u-ais. A central angle has been drawn subtending (cutting off) this arc. v radians radian 0.5 r r. One edge of inde card should be marked as shown here. Inde Card r r r u. Mark a fleible ruler (an inde card will do) with the scale shown on the u-ais. Then measure the arc on the circle opposite the angle marked radian. (See the illustration in Problem of the following Problem Set for ideas on how to do this.) Does the arc have length r? How does this fact tell ou that the angle measures radian?. Mark off two more radius-lengths on our fleible ruler starting from one end of the scale ou have alread marked. Then measure around the circumference of the circle from the positive u-ais to the terminal side of the angle marked. Based on this measurement and the definition of radians (page 0), what is the radian measure of angle?. What is the radian measure of one full revolution? An angle of 80? 90? 4. What is the degree measure of an angle of radian? Show how ou found our answer. 5. What did ou learn as a result of doing this eploration that ou did not know before? 0.5 r r Arc has length r, the radius, confirming that the angle is one radian. 00 Chapter 6: Applications of Trigonometric and Circular Functions

23 Ecerpt from an old Bablonian cuneiform tet r r Figure 6-4c a a The degree as a unit of angular measure came from ancient mathematicians, probabl Bablonians. It is assumed that the divided a revolution into 60 parts we call degrees because there were approimatel 60 das in a ear and the used the base-60 (seagesimal) number sstem. There is another wa to measure angles, called radian measure. This mathematicall more natural unit of angular measure is derived b wrapping a number line around the unit circle (a circle of radius unit) in a uv-coordinate sstem, as in Figure 6-4a. Each point on the number line corresponds to a point on the perimeter of the circle. v r u Figure 6-4a 5 4 v r If ou draw ras from the origin to the points,, and on the circle (right side of Figure 6-4a), the corresponding central angles have radian measures,, and, respectivel. But, ou ma ask, what happens if the same angle is in a larger circle? Would the same radian measure correspond to it? How would ou calculate the radian units measure in this case? Figures 6-4b and 6-4c answer these questions. Figure 6-4b shows an angle of unit measure, in radians, and the arcs it subtends (cuts rad off) on circles of radius unit and units. The arc r subtended on the unit circle has length unit. B the r properties of similar geometric figures, the arc Figure 6-4b subtended on the circle of radius has length units. So radian subtends an arc of length equal to the radius of the circle. For an angle measure, the arc length and the radius are proportional a r a r, as shown in Figure 6-4c, and their quotient is a unitless number that uniquel corresponds to and describes the angle. So, in general, the radian measure of an angle equals the length of the subtended arc divided b the radius.. Arc has length.8r, meaning that is.8 radians.. A full revolution is radians because the circumference of a circle is r. An angle of 80 is half of, or radians. An angle of 90 is _ 4 of, or _ radians. 4. One radian is of a full circle, so its degree measure is Answers will var. u Section 6-4: Radian Measure of Angles 0 Section Notes Section 6-4 introduces radian measures for angles. Measuring angles in radians makes it possible to appl trigonometric functions to real-life units of measure, such as time, that can be represented on the number line. Tr to do at least one of the eplorations for Section 6-4 so that students will understand that a radian is reall the arc length of a circle with a radius of one unit. The eplorations also help students understand the relationship between radians and degrees. If the do not complete one of these activities, students ma simpl memorize how to convert from degrees to radians without understanding how the units are related. Eplain that degrees and radians are two units for measuring angles, just as feet and meters are two units for measuring length. An angle measuring 0 radians is not the same size as an angle measuring 0 degrees (just as a length of 0 meters is not equal to a length of 0 feet) because the units are different. The relationship between radians and degrees can be used to convert measurements from one unit to the other. Because radians is equivalent to 80 degrees, the conversion factor is radians, a form of. Students should 80 degrees realize that multipling a number of degrees b this factor causes the degrees to cancel out, leaving a number of radians. 0 degrees radians 80 degrees 5 radians 6 Similarl, multipling a number of 80 degrees radians b, another form of, radians lets the radians cancel, leaving a number of degrees. If the units don t cancel, the student has the conversion factor upside down. Section 6-4: Radian Measure of Angles 0

24 Section Notes (continued) Another wa to make sure students can visualize a radian is to have each student hold up her or his hands, heel-to-heel, to form a one-radian angle. A radian is about 57 degrees. If the angle is much too big or too small, ou can correct them quickl. To help students develop a good understanding of radians, draw a large circle and mark several eact radian measures, including both integers and multiples of. On the same circle, show the decimal approimations for some of the radian measures, especiall,, and. Knowing decimal approimations for multiples of allows students to determine which quadrant an angle given in radians falls in. This helps students understand wh epressions like cos.487 and sin 5.9 represent negative quantities. If ou gave students a Trigonometric Ratios Table in Chapter 5, have them complete it b filling in the Radians column. EXAMPLE SOLUTION DEFINITION: Radian Measure of an Angle arc length radian measure radius For the work that follows, it is important to distinguish between the name of the angle and the measure of that angle. Measures of angle will be written this wa: is the name of the angle. m () is the degree measure of angle. m R () is the radian measure of angle. Because the circumference of a circle is r and because r for the unit circle is, the wrapped number line in Figure 6-4a divides the circle into units (a little more than si parts). So there are radians in a complete revolution. There are also 60 in a complete revolution. You can convert degrees to radians, or the other wa around, b setting up these proportions: m R () m () 60 or m () 80 m R () Solving for m R () and m (), respectivel, gives m R () m () and m () m R () These equations lead to a procedure for accomplishing the objective of this section. PROCEDURE: Radian Degree Conversion To find the radian measure of, multipl the degree measure b 80. To find the degree measure of, multipl the radian measure b 80. Convert 5 to radians. In order to keep the units straight, write 5 each quantit as a fraction with the proper = units. If ou have done the work correctl, certain units will cancel, leaving the proper units for the answer. 5 degrees m R () radians 80 degrees radians 4 Notes: eact value is called for, leave the answer as _ 4. If not, ou have the choice of writing the answer as a multiple of or converting to a decimal. dimensional analsis. You will use this procedure throughout our stud of mathematics. 0 Chapter 6: 0 Chapter 6: Applications of Trigonometric and Circular Functions

25 EXAMPLE SOLUTION EXAMPLE SOLUTION EXAMPLE 4 SOLUTION Convert 5.7 radians to degrees. 5.7 radians 80 degrees radians Find tan.7. Unless the argument of a trigonometric function has the degree smbol, it is assumed to be a measure in radians. (That is wh it has been important for ou to include the degree smbol until now.) Set our calculator to radian mode and enter tan.7. tan Find the radian measure and the degree measure of an angle whose sine is 0.. si n radian Set our calculator to radian mode. si n Set our calculator to degree mode. To check whether these answers are in fact equivalent, ou could convert one to the other. 80 degrees radian Use the alread in our radians calculator, without rounding. Radian Measures of Some Special Angles It will help ou later in calculus to be able to recall quickl the radian measures of certain special angles, such as those whose degree measures are multiples of 0 and 45. B the technique of Eample, 0 radian, or 6 revolution 45 radian, or 4 8 revolution If ou remember these two, ou can find others quickl b multiplication. For instance, 60 (/6) radians, or 6 revolution 90 (/6) or (/4) radians, or 4 revolution 80 6(/6) or 4(/4) radians, or revolution For 80, ou can simpl remember that a full revolution is radians, so half a revolution is radians. Differentiating Instruction Have students include radians in their journals, writing the definition in their own words. ELL students ma have been taught to use R to denote radian measure. Consider providing a table of trigonometric ratios or a unit circle for students to use as a reference. Alternativel, have students create their own unit circles. Be aware that students ma have learned conversions between degrees and radians using 60 and. Man languages use the same word for circumference and perimeter. ELL students ma interchange the words in English. Allow ELL students to do Problems and in pairs. The are language heav. Demonstrate for students the form ou want them to use for eact answers. Students ma have learned a variet of forms (across the class). It will take time for them to adjust. Section 6-4: 0 Section 6-4: Radian Measure of Angles 0

26 Additional Eploration Notes Eploration 6-4a can be completed as a whole-class activit. Problem 5 requires a protractor. It is important because it gives students an estimate of the number of degrees in one radian. Allow 0 5 minutes for this eploration. Technolog Notes Presentation Sketch: Radians Present.gsp at allows ou to animate a point along the radius of a circle to demonstrate the size of a oneradian angle measure. The sketch on the second page allows the point to animate all the wa around the circle and visuall demonstrates the approimate number of radians in 60 as well as the approimation 7. CAS Activit 6-4a: Inverse Trigonometric Functions in the Instructor s Resource Book has students eplore the relationships between the inverse trigonometric functions, and demonstrate that, although there are si inverse trigonometric functions, three of them are sufficient. Allow 5 0 minutes. Q. PROBLEM NOTES EXAMPLE 5 SOLUTION Problem Set 6-4 Reading Analsis From what ou have read in this section, what do ou consider to be the main idea? Is a radian large or small compared to a degree? How do ou find the radian measure of an angle if ou know its degree measure? How can ou remember that there are radians in a full revolution? Figure 6-4d shows the radian measures of some special first-quadrant angles. Figure 6-4e shows radian measures of larger angles that are _ 4, _, _ 4, and revolution. The bo summarizes this information. v, 90, 60 4, 45 6, 0 0, 0 u Quick Review v, 4 rev., rev., rev. u, 4 rev. Figure 6-4d Figure 6-4e PROPERTY: Radian Measures of Some Special Angles Degrees Radians /6 /4 / / Revolutions _ Find the eact value of sec 6. s e c sec 0 cos 0 6 / 0 60 sec = = cos Recall how to use the reference triangle to find the eact value of cos 0. hpotenuse adjacent Q. Sketch the graph of tan. Q. Sketch the graph of sec. Q. What is the first positive value of at which the graph of cot has a vertical asmptote? Q4. What is the first positive value of for which the graph of csc 0? Chapter 6: Q Q4. There is no value. Q Chapter 6: Applications of Trigonometric and Circular Functions

27 Q5. What is the eact value of tan 60? Q6. What transformation of function f is represented b g () f ()? Q7. What transformation of function f is represented b h () f (0)? Q8. Write the general equation for a quadratic function. Q ? Q0. The then part of the statement for a theorem is called the A. Converse B. Inverse C. Contrapositive D. Conclusion E. Hpothesis. Wrapping Function Problem: Figure 6-4f shows the unit circle in a uv-coordinate sstem. Suppose ou want to use the angle measure in radians as the independent variable. Imagine the -ais from an -coordinate sstem placed tangent to the circle. Its origin, 0, is at the point (u, v) (, 0). Then the -ais is wrapped around the circle. a. Show where the points,, and on the number line map onto the circle. b. On our sketch from part a, show angles of,, and radians in standard position. c. Eplain how the length of the arc of the unit circle subtended b a central angle of the circle is related to the radian measure of that angle. v Figure 6-4f 0 u. Arc Length and Angle Problem: As a result of the definition of radian, ou can calculate the arc length as the product of the angle in radians and the radius of the circle. Figure 6-4g shows arcs of three circles subtended b a central angle of. radians. The radii of the circles have lengths,, and cm. r. rad r arc r arc Figure 6-4g Section 6-4: arc a. How long would the arc of the -cm circle be if ou measured it with a fleible ruler? b. Find the lengths of the arcs on the -cm circle and the -cm circle using the properties of similar geometric figures. c. On a circle of radius r meters, what is the length of an arc that is subtended b an angle of. radians? d. How could ou quickl find the length a of an arc of a circle of radius r meters that is subtended b a central angle of radians? Write a formula representing the arc length. 05 Q5. Q6. -dilation of Q7. -dilation of 0. Q8. f () 5 a b c, a 0 Q Q0. D Problem introduces the net section and reinforces the ideas in Eploration 6-4. Be sure to assign this problem. A blackline master for Problem is available in the Instructor s Resource Book. a. b. v u r v u r c. The arc length on the unit circle equals the radian measure. Problem develops the general formula for finding the length of an arc given the angle measure and the radius of the circle. Be sure to assign this problem and review it in the homework discussion. a.. cm b..6 cm for r 5 cm;.9 cm for r 5 cm c..r m d. a 5 r Section 6-4: Radian Measure of Angles 05

28 Problem Notes (continued) Problems 0 can be done on a CAS using a Solve command and conversion ratios. Problems 0 provide skill practice in converting from degrees to eact radians Problems 4 provide skill practice in converting from degrees to approimate decimal forms of radians. Problems 5 4 provide skill practice in converting from radians to eact degree measures. Problems 5 0 provide skill practice in converting from radians to approimate decimal forms of degree measures. Problems 4 provide skill practice in finding approimate function values of angles in radians. Remind students to use the radian mode on their grapher. Problems 5 8 require students to find an angle in radians given a -value. Remind students to use radian mode on their grapher and that sin 0. doesn t mean. Discuss how students can sin 0. find cot and csc on their grapher using the tan and sin kes. Problems 9 48 provide skill practice in finding the eact function values for special angles in radians. Problems are eamples of trigonometric properties students will stud in Chapter 7. Problems review concepts from previous chapters. Remind students to put their graphers in degree mode for these problems. A blackline master for Problems 49 and 50 is available in the Instructor s Resource Book. See pages for answers to Problems 9 54 and CAS Problems. For Problems 0, find the eact radian measure of the angle (no decimals) For Problems 4, find the radian measure of the angle in decimal form For Problems 5 4, find the eact degree measure of the angle given in radians (no decimals). Use the most time-efficient method radian 6. radians 7. radian radian 9. radian 0. radians. radians. radians 4. radians radians For Problems 5 0, find the degree measure in decimal form of the angle given in radians radian radian 7..6 radians radians 9. radian 0. radians For Problems 4, find the function value (in decimal form) for the angle in radians.. sin 5. cos. tan(.) 4. sin 066 For Problems 5 8, find the radian measure (in decimal form) of the angle. 06 Chapter 6: ta n 5 = tan =5 5. si n ta n 5 7. co t 8. c sc.00 Additional CAS Problems. Put our CAS in radian mode and calculate t an _, cot 7_, csc 4, s in 8 95, sec 9_, and cos _ 4. Enter the epression in the first row of a table with the CAS result in the second row. There are si inverse trigonometric functions, but how man seem to be absolutel necessar? Eplain. 5 For Problems 9 44, find the eact value of the indicated function (no decimals). Note that because the degree sign is not used, the angle is assumed to be in radians. 9. sin 40. cos 4. tan 4. cot 6 4. sec 44. csc 4 For Problems 45 48, find the eact value of the epression (no decimals). 45. sin 6 cos 46. csc 6 sin co s si n 48. ta n s ec For Problems 49 and 50, write a particular equation for the sinusoid graphed For Problems 5 and 5, find the length of the side labeled in the right triangle For Problems 5 and 54, find the degree measure of angle in the right triangle cm cm 0 ft. Determine the measure of an angle whose radian measure numericall equals the square of its degree measure.. A sector of a circle with radius r is defined b an angle measuring radians. If the perimeter of the sector is 5 units and the area of the sector is 0 units, then what are the values of and r? 5 ft 06 Chapter 6: Applications of Trigonometric and Circular Functions

29 6-5 The normal human EKG (electrocadiogram) is periodic. Objective Circular Functions In man real-world situations, the independent variable of a periodic function is time or distance, with no angle evident. For instance, the normal dail high temperature varies periodicall with the da of the ear. In this section ou will learn about circular functions, periodic functions whose independent variable is a real number without an units. These functions, as ou will see, are identical to trigonometric functions in ever wa ecept for their argument. Circular functions are more appropriate for real-world applications. The also have some advantages in later courses in calculus, for which this course is preparing ou. Two ccles of the graph of the parent cosine function are completed in 70 (Figure 6-5a, left) or in 4 units (Figure 6-5a, right), because 4 radians correspond to two revolutions. cos Figure 6-5a cos To see how the independent variable can represent a real number, imagine the -ais from an -coordinate sstem lifted out and placed verticall tangent to the unit circle in a uv-coordinate sstem with its origin at the point (u, v) (, 0), as on the left side in Figure 6-5b. Then wrap the -ais around the unit circle. As shown on the right side in Figure 6-5b, maps onto an angle of radian, maps onto radians, maps onto radians, and so on. v -ais rad u 0 Figure 6-5b Wrapped -ais v 4 rad u 0 Arc of length Section 6-5: 07 Section 6-5 Class Time da PLANNING Homework Assignment RA, Q Q0, Problems, 6, 7, 0,, 4, 5, 9,, 5, 7,, 7, 9, 44, (45, 46, 48), 49 Teaching Resources Eploration 6-5a: Circular Function Parent Graphs Supplementar Problems Technolog Resources Sinusoid Translation (Problem 45) Sinusoid Dilation (Problem 46) The Inequalit sin tan (Problem 48) Presentation Sketch: Circular Functions Present.gsp Presentation Sketch: Circular Transforms Present.gsp Presentation Sketch: Sine Challenge Present.gsp Eploration 6-5a: Circular Function Parent Graphs Activit: Trigonometr Tracers Activit: Transformations of Circular Functions TEACHING Important Terms and Concepts Circular functions Standard position Section 6-5: Circular Functions 07

30 Section Notes Section 6-5 introduces circular functions. Circular functions are identical to trigonometric functions ecept that their arguments are real numbers without units rather than degrees. The radian, introduced in the previous section, provides the link between trigonometric functions and circular functions. For Eample on page 09 and the homework problems, ou ma need to remind students that the period for the parent tangent graph and the parent cotangent graph is, not. Summarize the section with the chart at the bottom of this page to help students learn the clues (words and smbols) that determine whether degree or radian mode should be used. Note: Although there is no smbol for radians is a number without an units students can use as a clue for radians. Differentiating Instruction Students ma not be accustomed to the convention of using Greek letter arguments to represent degrees and Roman letters to represent radians. Be sure to point out this distinction and provide several eamples to help students become comfortable with this distinction. Consider allowing ELL students to do Problems in pairs and providing them with the answers. These problems contain challenging language and important concepts. Alternativel, work through these problems as a class. Eploration Notes Eploration 6-5a requires students to investigate graphs of parent trigonometric and circular functions. 08 Chapter 6: Problems ask students to sketch the parent sine, cosine, and tangent graphs from 0 through 70. Problems 4 6 have students graph the same three functions in radian mode. Problem 7 asks students to relate the periods of the graphs to degrees and radians. Problem 8 asks students to write an equation for a given sinusoid (the same sinusoid as in Eample ). You can use this eploration to introduce the section or as a quiz after the section is completed. The distance on the -ais is equal to the arc length on the unit circle. This arc length is equal to the radian measure for the corresponding angle. Thus the functions sin and cos for a number on the -ais are the same as the sine and cosine of an angle of radians. Figure 6-5c shows an arc of length on the unit circle, with the corresponding angle. The arc is in standard position on the unit circle, with its initial point at (, 0) and its terminal point at (u, v). The sine and cosine of are defined in the same wa as for the trigonometric functions: cos horizontal coordinate radius sin vertical coordinate radius u u v v The name circular function comes from the fact that equals the length of an arc on the unit circle. The other four circular functions are defined as ratios of sine and cosine. DEFINITION: Circular Functions If (u, v) is the terminal point of an arc of length in standard position on the unit circle, then the circular functions of are defined as sin v cos u tan cos sin v u cot cos sin sec cos u csc sin Circular functions are equivalent to trigonometric functions in radians. This equivalenc provides an opportunit to epand the concept of trigonometric functions. You have seen trigonometric functions first defined using the angles of a right triangle and later epanded to include all angles. From now on, the concept of trigonometric functions includes circular functions, and the functions can have both degrees and radians as arguments. The wa the two kinds of trigonometric functions are distinguished is b their arguments. If the argument is measured in degrees, Greek letters represent them (for eample, sin ). If the argument is measured in radians, the functions are represented b letters from the Roman alphabet (for eample, sin ). u v v rad (cos, sin ) (u, v) v sin u cos Figure 6-5c If ou use it as an introduction, allow 5 minutes. If ou use it as a quiz, allow less time and consider asking students to do their work without their graphers. Degrees Radians Phrase Trigonometric Circular Variable Smbol None () arc u 08 Chapter 6: Applications of Trigonometric and Circular Functions

31 4 EXAMPLE SOLUTION Figure 6-5d EXAMPLE SOLUTION Technolog Notes Plot the graph of 4 cos 5 on our grapher, in radian mode. Find the period graphicall and algebraicall. Compare our results. Figure 6-5d shows the graph. Tracing the graph, ou find that the first high point beond 0 is between.5 and.. So graphicall the period is between.5 and.. To find the period algebraicall, recall that the 5 in the argument of the cosine function is the reciprocal of the horizontal dilation. The period of the parent cosine function is, because there are radians in a complete revolution. Thus the period of the given function is ( ) The answer found graphicall is close to this eact answer. Note: Using the MAXIMUM feature of our grapher confirms that the high point is at Find a particular equation for the sinusoid function graphed in Figure 6-5e. Notice that the horizontal ais is labeled, not, indicating that the angle is measured in radians. Confirm our answer b plotting the equation on our grapher. 0 Figure 6-5e C A cos B( D), so C. a high point, so A. Write the general sinsoidal equation, using instead of. Find A, B, C, and D using information from the graph. From one high point to the net is. 0 5 or, so B. B is the reciprocal of the 5 horizontal dilation. parent function cos ), s o D. cos ( ) 5 Cosine starts a ccle at a high point. Write the particular equation. Plotting this equation in radian mode confirms that it is correct. Problem 45: The Sinusoid Translation Problem asks students to eplore translations of sine and cosine graphs with the help of the Dnamic Precalculus Eploration at Section 6-5: 09 Problem 46: The Sinusoid Dilation Problem asks students to eplore, with the help of the Dnamic Precalculus Eploration at dilations of the sine graph and the effects the dilations have on the period of the graph. Problem 48: The Inequalit sin tan Problem asks students to eplore the inequalit sin tan with the help of an interactive sketch in the Dnamic Precalculus Eploration at Presentation Sketch: Circular Functions Present.gsp at demonstrates all si trigonometric functions as lengths of certain legs of a triangle constructed around the unit circle. This sketch could serve as an introduction to the activit Transformations of Circular Functions, in which students are asked to construct these triangles themselves. Presentation Sketch: Circular Transforms Present.gsp at animates a point on the unit circle to show the graphs of sine, cosine, and tangent as well as a horizontal dilation of these three graphs. This could serve as an introduction to the activit Transformations of Circular Functions. Presentation Sketch: Sine Challenge Present.gsp at plots a point and asks for a dilation of the sine function that passes through the point. You ma wish to have students eperiment with the sketch on their own. Eploration 6-5a: Circular Function Parent Graphs in the Instructor s Resource Book can be enhanced b using Sketchpad. Students can graph in either radians or degrees b changing the Preferences in the Edit menu. Section 6-5: Circular Functions 09

32 Technolog Notes (continued) Activit: Trigonometr Tracers in the Instructor s Resource Book guides students through a Sketchpad construction of the graphs of sine, cosine, and tangent based on the coordinates of a point on a unit circle. Allow 0 minutes. Activit: Transformations of Circular Functions in the Instructor s Resource Book has students start with a base sketch and build it up to be similar to the presentation sketch Circular Transforms Present.gsp, mentioned earlier. It is an etended activit emphasizing the effect on period and amplitude of dilations of the sine and cosine graphs. Allow 45 minutes. CAS Suggestions Remember that Boolean logic can be used to confirm the computation of the period of a trigonometric function. PROBLEM NOTES A Supplementar Problem on reflection of the sine function is available for this section at keonline. Q. Q. 608 Q Q Q Q Q7. 78 Q8. 7 Q9. 5 h Q Problems 4 review ideas from Section units. 6 units. units 4. 4 units EXAMPLE 6 Figure 6-5f 0 Chapter 6: SOLUTION Problem Set 6-5 Reading Analsis Sketch the graph of tan 6. In order to graph the function, ou need to identif its period, the locations of its inflection points, and its asmptotes. Period 6 6 Horizontal dilation is the reciprocal of ; the period of 6 the parent tangent function is. For this function, the points of inflection are also the -intercepts, or the points where the value of the function equals zero. So 0,,, , 6,,... From what ou have read in this section, what do ou consider to be the main idea? As defined in this tet, what are the differences and the similarities between a circular function and a trigonometric function? How do angle measures in radians link the circular functions to the trigonometric functions? 5min Quick Review Q. How man radians are in 80? Q. How man degrees are in radians? Q. How man degrees are in radian? Q4. How man radians are in 4? Q5. Find sin 47. Q6. Find sin 47. Q7. Find the period of 4 cos 5( 6 ). Q8. Find the upper bound for for the sinusoid in Problem Q7. In Problems 9 check that students understand that on a unit circle the length of an arc is equal to the radian measure of the corresponding angle. Students are often puzzled b these problems because the seem too simple. Do not provide an eample for Problems 9. Allow students to think the problems through. 9. units 0. units. units..467 units Asmptotes are at values where the function is undefined. So 6,,, 5,...,, 9, 5,... Recall that halfwa between a point of inflection and an asmptote the tangent equals or. The graph in Figure 6-5f illustrates these features. Note that in the graphs of circular functions the number appears either in the equation as a coefficient of or in the graph as a scale mark on the -ais. Q9. How long does it take ou to go 00 mi at an average speed of 60 mi/h? Q0. Write 5% as a decimal. For Problems 4, find the eact arc length on the unit circle subtended b the given angle (no decimals) For Problems 5 8, find the eact degree measure of the angle that subtends the given arc length on the unit circle. 5. units 6. 6 unit 7. unit 8. 4 units For Problems 9, find the eact arc length on the unit circle subtended b the given angle in radians. 9. radians 0. radians. radians..467 radians Problems 4 are trivial for students using a CAS undefined 0 Chapter 6: Applications of Trigonometric and Circular Functions

33 For Problems 6, evaluate the circular function in decimal form.. tan 4. sin 5. sec 6. cot 4 For Problems 7 0, find the inverse circular function in decimal form. 7. co s ta n.4 9. c sc 5 0. s ec 9 For Problems 4, find the eact value of the circular function (no decimals).. sin. t an 6. cos 4 4. csc For Problems 5 8, find the period, amplitude, phase displacement, and sinusoidal ais location. Use these features to sketch the graph. Confirm our graph b plotting the sinusoids on our grapher. 5. cos ( 4) sin ( ) Period 5 8; Amplitude 5 6; Phase displacement 5 ; Sinusoidal ais Period 5 6; Amplitude 5 4; Phase displacement 5 ; Sinusoidal ais Period 5 4; Asmptotes at 4n; Points of inflection at 4n 7. 6 sin ( ) cos ( ) For Problems 9, find the period, asmptotes, and critical points or points of inflection, and then sketch the graph. 9. cot 0. tan 4. sec. csc For Problems 4, find a particular equation for the circular function graphed Period 5 ; Asmptotes at 4 n; Points of inflection at n Problems 5 provide practice in graphing a circular function given its equation. Watch for students who use instead of as the period of the parent tangent and cotangent graphs. 5. Period 5 0; Amplitude 5 ; Phase displacement 5 4; Sinusoidal ais Section 6-5: 6. Period 5 ; Amplitude 5 5; Phase displacement 5 ; Sinusoidal ais Problems 4 require students to write the equation of a circular function given its graph. Problems 4 and 4 provide onl a portion of one ccle cos ( ) cos cos 5 ( 5) cos ( ) csc cot 4 See page 005 for answers to Problems and. Section 6-5: Circular Functions

34 Problem Notes (continued) 9. 5 tan sec 4. z 5 8 sin 5(t 0.7) 4. E cos 800 (r 00) Problems 4 and 44 prepare students for the net section. Definitel assign one of these problems. 4. z (0.4) ; z (50) ; z (50) is below the sinusoidal ais. 44. E(4) ; E(0,000) ; E(0,000) is above the sinusoidal ais. The solution to Problem 45a could be confirmed with Boolean algebra on a CAS. In Problem 45b, a CAS automaticall reduces the given epression to cos, suggesting that the horizontall translated graph coincides with the original graph. 45a. Horizontal translation of sin 5 cos 45b. Horizontal translation of ; the graph would coincide with itself and appear unchanged. 45c. or, or an multiple of 45d. A horizontal translation b a multiple of results in a graph that coincides with itself. The period of the sine function is. 45e. Answers will var. As k increases, the graph moves to the right. Problem 46 has students investigate horizontal stretches of sinusoids using the Sinusoid Translation eploration at Students should alread feel comfortable with these transformations, but this problem gives them an opportunit to eamine them from a different perspective. ; For the sinusoid in Problem 4, find the value of z at t 0.4 on the graph. If the graph is etended to t 50, is the point on the graph above or below the sinusoidal ais? How far above or below? 44. For the sinusoid in Problem 4, find the value of E at r 4 on the graph. If the graph is etended to r 0,000, is the point on the graph above or below the sinusoidal ais? How far above or below? Chapter 6: z E t a. Because the length of the hpotenuse 5 the radius of the circle 5, v 5 sin 5 radius of circle 5 v 5 v, v and 5 sin 5 5 v. radius of circle 5 v 900 r 45. Sinusoid Translation Problem: Figure 6-5g shows the graphs of cos (dashed) and sin (solid). Note that the graphs are congruent to each other (if superimposed, the coincide), differing onl in horizontal translation. Figure 6-5g 4 a. What translation would make the cosine graph coincide with the sine graph? Complete the equation: sin cos(? ). b. Let cos( ). What effect does this translation have on the cosine graph? c. Name a positive and a negative translation that would make the sine graph coincide with itself. d. Eplain wh sin( n) sin for an integer n. How is the related to the sine function? e. Using dnamic geometr software such as The Geometer s Sketchpad, plot two sinusoids with different colors illustrating the concept of this problem, or use the Sinusoid Translation eploration at One sinusoid should be cos and the other cos( k), where k is a slider or parameter with values between and. Describe what happens to the transformed graph as k varies. 46. Sinusoid Dilation Problem: Figure 6-5h shows the unit circle in a uv-coordinate sstem with angles of measure and radians. The uv-coordinate sstem is superimposed on an -coordinate sstem with sinusoids sin (dashed) and an image graph sin (solid). 46b. Answers will var. The second angle measure is double the first, but the moving points on the sinusoids alwas have the same -values. Chapter 6: Applications of Trigonometric and Circular Functions

35 v or (, ) (, ) (u, v ) (u, v ) u or 4 5 Figure 6-5h a. Eplain wh the value of v for each angle is equal to the value of for the corresponding sinusoid. b. Create Figure 6-5h with dnamic geometr software such as Sketchpad, or go to and use the Sinusoid Dilation eploration. Show the whole unit circle, and etend the -ais to 7. Use a slider or parameter to var the value of. Is the second angle measure double the first one as varies? Do the moving points on the two sinusoids have the same value of? c. Replace the in sin with a variable factor, k. Use a slider or parameter to var k. What happens to the period of the (solid) image graph as k increases? As k decreases? 47. Circular Function Comprehension Problem: For circular functions such as cos, the independent variable,, represents the length of an arc of the unit circle. For other functions ou have studied, such as the quadratic function a b c, the independent variable,, stands for a distance along a horizontal number line, the -ais. a. Eplain how the concept of wrapping the -ais around the unit circle links the two kinds of functions. b. Eplain how angle measures in radians link the circular functions to the trigonometric functions. 46c. As k increases, the period decreases, and vice versa. The period is alwas k. 48. The Inequalit sin tan Problem: In this problem ou will eamine the inequalit sin tan for 0. Figure 6-5i shows angle AOB in standard position, with subtended arc AB of length on the unit circle. B D O C A Figure 6-5i a. Based on the definition of radians, eplain wh is also the radian measure of angle AOB. b. Based on the definitions of sine and tangent, eplain wh BC and AD equal sin and tan, respectivel. c. From Figure 6-5i it appears that sin tan. Make a table of values to show numericall that this inequalit is true even for values of ver close to zero. d. Construct Figure 6-5i with dnamic geometr software such as Sketchpad, or go to and use the Inequalit sin tan eploration. On our sketch, displa the values of and the ratios (sin )/ and (tan )/. What do ou notice about the relative sizes of these values when angle AOB is in the first quadrant? What value do the two ratios seem to approach as angle AOB gets close to zero? 49. Journal Problem: Update our journal with things ou have learned about the relationship between trigonometric functions and circular functions. Section 6-5: Problem 47 is a good journal problem. 47a. Wrapping the -ais around the unit circle converts distances along the -ais to arc lengths, and vice-versa. In particular, it shows that a circular function s independent variable (arc length) is the same as a distance along the -ais. So for both tpes of functions, the independent variable is a distance along the -ais. 47b. A radian measure corresponds to an angle measure, using m R () 5 m(), but because a radian measure 80 is a pure number, it can represent something other than an angle in an application problem. Problem 48 makes an ecellent group activit. Sin is the length of the vertical segment BC, and tan is the length of the vertical segment if the radius is etended to a point (, tan ). 48a. m R (AOB) 5 arc radius b. The circle is a unit circle. Hence BC 5 sin, and AD 5 AD 5 tan. 48c. OA sin tan d. sin, but approaches as approaches 0; tan, but also approaches as approaches Journal entries will var. Additional CAS Problems. A sector of a circle has an area that is numericall 5 units larger than the length of its arc. What is the eact value of the radius of the circle if the radius is unit shorter than the arc length?. What is the eact value of the radius of a circle for which the arclength of a sector is equivalent to the radian angle measure of the sector?. What is the eact value of the radius of a circle for which the area of a sector of the circle is equivalent to the radian angle measure of the sector? See page 005 for answers to CAS Problems. Section 6-5: Circular Functions

36 Section 6-6 Class Time da PLANNING Homework Assignment RA, Q Q0, Problems odd Teaching Resources Eploration 6-6a: Sinusoids, Given, Find Numericall Eploration 6-6b: Given, Find Algebraicall Supplementar Problems 6-6 Objective Inverse Circular Relations: Given, Find A major reason for finding the particular equation for a sinusoid is to use it to evaluate for a given -value or to calculate when ou are given. Functions are used this wa to make predictions in the real world. For instance, ou can epress the time of sunrise as a function of the da of the ear. With this equation, ou can predict the time of sunrise on a given da b simpl evaluating the epression. Predicting the da(s) on which the Sun rises at a given time is more complicated. In this section ou will learn graphical, numerical, and algebraic was to fi n d for a given value of. Radar speed guns use inverse relations to calculate the speed of a car from time measurements. value of or Technolog Resources Eploration 6-6a: Sinusoids, Given, Find Numericall TEACHING Important Terms and Concepts Inverse circular relations Inverse cosine function Inverse cosine relation Arccosine Arccos General solution Principal value Inverse circular function Section Notes Section 6-6 introduces students to the inverse cosine relation, or arccosine, and discusses the general solution for the arccosine of a number. This section is critical for understanding how to solve equations involving circular functions and must not be omitted. The real-world problems in Section 6-7 require the skills learned in this section. To introduce this section numericall, have students find the following cosine values (radian mode).. cos The Inverse Cosine Relation The smbol co s 0. means the inverse cosine function evaluated at 0., a particular arc or angle whose cosine is 0.. B calculator, in radian mode, co s The inverse cosine relation includes all arcs or cos 0. angles whose cosine is a given number. The term that ou ll use in this tet is arccosine, Figure 6-6a abbreviated arccos. So arccos 0. means an arc or angle whose cosine is 0., not just the function value. Figure 6-6a shows that both and have cosines equal to 0.. So is also a value of arccos 0.. Th e general solution for the arccosine of a number is written this wa: 4 Chapter 6: Applications of Trigonometric and Circular Functions 5? Answer: 0.5. cos 5? Answer: 0.5. cos 7 5? Answer: cos 8 5? Answer: 0.5 Then have them find the value of the inverse cosine function, cos 0.5, using a grapher. The answer is the single value , which equals onl. The inverse cosine u 0. arccos 0. c os 0. n General solution for arccos 0.. where n stands for an integer. The sign tells ou that both the value from the calculator and its opposite are values of arccos 0.. The n tells ou that an arc that is an integer number of revolutions added to these values is also a value of arccos 0.. If n is a negative integer, a number of revolutions is being subtracted from these values. Note that there are infinitel man such values. The arcsine and arctangent relations will be defined in Section 7-4 in connection with solving more general equations. relation, arccos 0.5, has an infinite number of values, including all four of the values shown in Problems 4 to the left. Ask students to find the difference between the angles in Problems and, and between the angles in Problems and 4. (The difference is, an angle of one complete revolution.) Then ask them wh the difference between the angles in Problems and is not also. (There are two angles in each revolution that have equal cosines.) v cos u 4 Chapter 6: Applications of Trigonometric and Circular Functions

37 EXAMPLE SOLUTION DEFINITION: Arccosine, the Inverse Cosine Relation arccos c os n or arccos c os 60 n where n is an integer Verball: Inverse cosines come in opposite pairs with all their coterminal angles. Note: The function value co s is called the principal value of the inverse cosine relation. This is the value the calculator is programmed to give. In Section 7-6, ou will learn wh certain quadrants are picked for these inverse function values. Find the first five positive values of arccos(0.). Assume that the inverse circular function is being asked for. arccos (0.) c os (0.) n n B calculator , , Use co s (0.). or , Use c o s (0.) , , or , , , , Arrange in , ascending order. Note: Do not round the value of co s (0.) before adding the multiples of. Finding When You Know Figure 6-6b shows a sinusoid with a horizontal line drawn at 5. The horizontal line cuts the part of the sinusoid shown at si different points. Each point corresponds to a value of for which 5. The net eamples show how to find the values of b three methods. 6 In Eample, point out the difference between the 5 sign and the sign. Eample 4 on page 6 shows how to find a general solution of the equation 9 7 cos ( 4) 5 5. When discussing this eample, emphasize that after the general solution is substituted for arccosine, the coefficient must be distributed over both terms. In solving equations like this, students often forget to distribute the coefficient over the n. Point out that the coefficient of n in the general solution should match the period of the graph. Once a general solution is found, particular solutions can be obtained b substituting integer values for n. A grapher table is an efficient wa to find particular solutions, using these functions: f () 5 4 /() cos (4/7) f () 5 4 /() cos (4/7) (or n) f ( ) f ( ) EXAMPLE SOLUTION Figure 6-6b Find graphicall the si values of for which 5 for the sinusoid in Figure 6-6b. On the graph, draw vertical lines from the intersection points down to the -ais (Figure 6-6b). The values are 4.5, 0.5, 8.5,.5,.5, 5.5 It is important that students understand that the definition of arccosine given in the tet requires that n stand for an integer and that the arccosine of a number is a collection of angle measurements. Follow up graphicall b having students work Eploration 6-6a. Algebraic solutions for as presented in Eploration 6-6b can be done the same da in longer blockscheduling periods. Section 6-6: 5 The eamples in Section 6-6 use graphical, numerical, and algebraic methods to find the -values of a sinusoid that correspond to a par ticular -value. Be sure to discuss the numerical method in detail so students understand that the must find both of the adjacent -values and then find additional answers b adding multiples of the period to each of these values. Have students store each of the adjacent -values in the memor of their calculators. Before allowing students to use grapher tables to find solutions, make sure the have had sufficient practice finding solutions using the paper-and-pencil method and that the demonstrate an understanding of the idea behind selecting values of n. Section 6-6: Inverse Circular Relations: Given, Find 5

38 Differentiating Instruction Point out the mathematical meanings of particular, principal, and supplementar. Bilingual dictionaries ma give onl the standard English d e fi n it ion s. There are several different was to write n and 60 n. Students ma have learned to write this as k or z and 60 k or 60 z. Be aware that some students who have studied trigonometr (to varing levels) ma have been taught to leave answers (such as those in Eample 4) in fraction form. Confirm that students understand our epectation regarding how to write their answers. The Reading Analsis ma be confusing for students who have studied trigonometr from a different approach. Eploration Notes Eploration 6-6a asks questions based on a given sinusoidal graph. You can use this eploration instead of doing the eamples with our class. The eamples can reinforce what students learned through the eploration. Students find values graphicall, numericall, and algebraicall. This eploration can be used as a small-group introduction to the section, as a dail quiz after the section is completed, or as a review sheet to be assigned later in the chapter. If ou use the eploration to introduce the section, make sure students have written a correct equation for Problem before the work on the other problems. Allow about 0 minutes for the eploration. Eploration 6-6b has students find the same -values for the same function as in Eploration 6-6a, but using algebraic methods rather than graphical ones. Allow 5 0 minutes for this eploration. 6 Chapter 6: Technolog Notes EXAMPLE SOLUTION EXAMPLE 4 SOLUTION Find numericall the si values of in Eample. Show that the answers agree with those found graphicall in Eample. f 9 7 cos () ( 4) f () and () () () () These answers agree with the answers found graphicall in Eample. Note that the sign is used for answers found numericall because the solver or intersect feature on most calculators gives onl approimate answers. Find algebraicall (b calculation) the si values of in Eample. Show that the answers agree with those in Eamples and. 9 7 cos ( 4) 5 cos ( 4) 4 7 Eploration 6-6a: Sinusoids, Given, Find Numericall in the Instructor s Resource Book can be done with the aid of Sketchpad or Fathom. CAS Suggestions Eample can be solved efficientl using the command Solve(cos = 0., ). Note the return of the smbol n# (where # is an integer) in the CAS result. ( 4) arccos arccos c o s 4 7 n 4 co s 4 7 n n n or n , , , , , Let n be 0,,. These answers agree with the graphical and numerical solutions in Eamples and. Write the particular equation using the techniques of Section 6-5. Plot a horizontal line at 5. Use the intersect or solver feature on our grapher to find two adjacent -values. Add multiples of the period to find other -values. Set the two functions equal to each other. Simplif the equation b isolating the cosine epression (start peeling constants awa from ). Take the arccosine of both sides. Rearrange the equation to isolate (finish peeling constants awa from ). Substitute for arccosine. Distribute the over both terms. 6 Chapter 6: Applications of Trigonometric and Circular Functions

39 Problem Set 6-6 Reading Analsis Notes: From what ou have read in this section, what do ou consider to be the main idea? Wh does the arccosine of a number have more than one value while co s of that number has onl one value? What do ou have to do to the inverse cosine value ou get on our calculator in order to find other values of arccosine? Eplain the sentence Inverse cosines come in opposite pairs with all their coterminal angles that appears in the definition bo for arccosine. Quick Review Q. What is the period of the circular function cos 4? Q. What is the period of the trigonometric function cos 4? Q. How man degrees are in 6 r a d i a n? Q 4. How man radians are in 45? Q5. Sketch the graph of sin. Q6. Sketch the graph of csc. Q7. Find the smaller acute angle in a right triangle with legs of length mi and 7 mi. Q8. 9 is the equation of a(n)?. Q9. What is the general equation for an eponential function? Q0. Functions that repeat themselves at regular intervals are called? functions. 5min n, the is the period. The n in the general solution for means that ou need to add multiples of the period to the values of ou find for the inverse function. n and n into our grapher and make a table of values. For most graphers ou will have to use in place of n. time numericall. To list the values simultaneousl, convert the previous answer line into list format and use a condition at the end of the line for the random integer n9 to assume the values 0,,, and the first four non-negative integers. The command should look like this: {*n9* , *n9* } n9 = {0,,, } A similar approach can be used in Eamples and 4. For Problems 4, find the first five positive values of the inverse circular relation.. arccos 0.9. arccos 0.4. arccos(0.) 4. arccos(0.5) For the circular sinusoids graphed in Problems 5 0, a. Estimate graphicall the -values shown for the indicated -value. b. Find a particular equation for the sinusoid. c. Find the -values in part a numericall, using the equation from part b. d. Find the -values in part a algebraicall. e. Find the first value of greater than 00 for which the sinusoid has the given -value Section 6-6: To find a specific intersection among the infinite solutions to an inverse trigonometric relation, it is helpful to state constraints after the Solve command. The graph in Problem 5 seems to have, among others, an intersection with 5 6 somewhere between 5 0 and 5. To find this intersection, students can enter Solve( + 5 cos(/0( )) = 6, ) 0 < < into their CAS. PROBLEM NOTES Supplementar Problems designed to introduce arcsine and arctangent are available at keonline. Q. Q. 908 Q. 08 Q4. 4 Q7. tan Q8. Circle of radius and center (0, 0) Q9. 5 a b, a 0, b 0 Q0. Periodic Problems 4 provide students with practice solving inverse circular relation equations. Because no method is specified, students ma find the solutions numericall, graphicall, or algebraicall. Problems 4 can be solved on a CAS using the method suggested in the CAS Suggestions. Parts c of Problems 5 can be approached as described in the CAS Suggestions. Using appropriate domain restrictions, students can calculate all of the solutions simultaneousl. Problems 5 0 require students to write equations for sinusoids and use graphical, numerical, and algebraic methods to find -values corresponding to a particular -value. These problems are similar to the eamples and to Eploration 6-6a. In part a, students should estimate the graphical solutions from the graph and not use the intersect feature on their grapher. One wa to answer part e of Problems 5 0 is to restrict to values between 00 and 00 one period. Based on the ccle of the graph, there should be two answers, both of which are calculated when using a CAS. See pages for answers to Problems Q5, Q6, and 6. Section 6-6: Inverse Circular Relations: Given, Find 7

40 Problem Notes (continued) 7a..9, 0.5,.,.5, 5. 7b. 5 4 cos ( 0.) 7c , ,.9..., , d , ,.9..., , e a..,.5, 4.9 8b. 5 cos ( 0.7) 8c , , d , , e a. 0.6,.4, 5.4,.6,.4 9b. 5 cos ( ) 8 9c , , , , d , , , , e a. 0, 6,, 6, 4 0b. 5 5 cos ( ) 0c , , , , d , , , , e Problems and involve inverse trigonometric functions rather than inverse circular functions. In the homework discussion, point out the similarities and differences between Problems and, and Problems 5 0. a. 0, 70, 0 b cos ( 60) c , , d , , Chapter 6: For the trigonometric sinusoids graphed in Problems and, a. Estimate graphicall the first three positive values of for the indicated -value. b. Find a particular equation for the sinusoid. c. Find the -values in part a numericall, using the equation from part b. d. Find the -values in part a algebraicall. a. 5, b. 5 4 cos 9 ( 0) c...., , 0... d. 5..., , Figure 6-6c shows the graph of the parent cosine function, cos. a. Find algebraicall the si values of shown on the graph for which cos 0.9. b. Find algebraicall the first value of greater than 00 for which cos Figure 6-6c Problem asks students to use algebra to find solutions to cos Be sure to assign this problem because it requires students to find a ver large -value not shown on the graph. a , , , , , b Chapter 6: Applications of Trigonometric and Circular Functions

41 6-7 Objective Sinusoidal Functions as Mathematical Models A chemotherap treatment destros 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 ever three weeks, then the red cell count resembles a periodic function of time (Figure 6-7a). If such a function is regular enough, ou can use a sinusoidal function as a mathematical model. In this section ou ll start with a verbal description of a periodic phenomenon, interpret it graphicall, find an algebraic equation from the graph, and use the equation to calculate numerical answers. In this eploration ou will use sinusoids to predict events in the real world. EXPLORATION 6-7: C h e m o t h e r a p P r o b l e m Section 6-7: Red cell count 6 Time (wk) Figure 6-7a. Figure 6-7a shows the red blood cell count. For what interval of times around 9 weeks for a patient taking chemotherap treatments will the patient feel good? For how man das each weeks. Suppose that a treatment is does the good feeling last? Show how ou got given at time t 0 weeks, when the count our answers. is c 800 units. The count drops to a low of 4. What did ou learn as a result of doing this 00, and then rises back to 800 when t, at eploration that ou did not know before? which time the net treatment is given. Write the equation for this sinusoidal function.. Th e patient feels good if the count is above 700. Use the equation in Problem to predict the count 7 das after the treatment, and thus conclude whether she has started feeling good again at this time. Eploration Notes Eploration 6-7 shows a chemotherap model to predict events in the real world. Note that the topic might be sensitive for students who have had a friend or relative suffer from cancer. Check student progress after each question, and make sure everone gets the correct answer before moving on to the net question. Note that students must convert their answers between weeks and das. 9 An alternate version of this eploration can be found in the Instructor s Resource Book. Allow 5 0 minutes. See page 0 for notes on additional eplorations.. c cos , so the patient is not feeling good. 4. Answers will var. 9 Section 6-7 Class Time das PLANNING Homework Assignment Da : RA, Q Q0, Problems, Da : Problems 4, 5, 8,, (4, 5) Teaching Resources Eploration 6-7: Chemotherap Problem Eploration 6-7a: Oil Well Problem Supplementar Problems Technolog Resources Waterwheel Eploration 6-7: Chemotherap Problem Eploration 6-7a: Oil Well Problem CAS Activit 6-7a: Epicenter of an Earthquake TEACHING Important Terms and Concepts Mathematical model Section Notes Section 6-7 contains a wide variet of real-world problems that illustrate how common sinusoids are in real life. It is recommended that ou spend two das on this section. On the first da, do Eploration 6-7 as a whole-class activit. Then have students work in small groups on Problem. This problem is ver similar to the Chemotherap Problem. On the second da, discuss the homework and have students work on other problems. See page 006 for answers to Problems and. Section 6-7: Sinusoidal Functions as Mathematical Models 9

42 Section Notes (continued) To answer part d of Eample correctl, students must analze the graph. If students tr to find the solution without looking at the graph, the are likel to give the first positive t-value that gives a d-value of zero. However, this solution is the time when the point first enters the water, not the time when it emerges from the water. Some of the problems in the problem set make ecellent projects. For eample, ou might divide the class into groups of three to five students and have each group make a video demonstrating the situation described in one of the problems. Ask students to use real stopwatches, tape measures, and other measuring devices to accuratel measure times and distances. From their collected measure ments, students can write a particular equation and use it to find -values given -values and to find -values given -values. A ke point to emphasize in this section is that almost all real-world sinusoid problems involve working with real numbers (i.e., radians) and that students need to have their grapher in radian mode. Students must learn to decide whether to use radians or degrees based on the contet of a problem, not just on whether the problem includes or or. Differentiating Instruction This section contains a significant amount of language challenging to ELL students. Be prepared to help them understand words and phrases such as waterwheel, emerge, chemotherap, red blood cell count, and so on. Have ELL students do the problems in pairs. The will benefit from spending a third da on this section; however, the language will still present a significant challenge. Waterwheel Water surface d (ft) 6 EXAMPLE 7 ft 6 ft Figure 6-7b d 7 Figure 6-7c d t? d 0 Figure 6-7d 0 Chapter 6: Rotation P SOLUTION t 7.5 d? t (s) Provide ELL students with a sense of the cultural significance of Mark Twain (Samuel Clemens) before assigning Problem. t Waterwheel Problem: Suppose that the waterwheel in Figure 6-7b rotates at 6 revolutions per minute (rev/min). Two seconds after ou start a stopwatch, point P on the rim of the wheel is at its greatest height, d ft, above the surface of the water. The center of the waterwheel is 6 ft above the surface. a. Sketch the graph of d as a function of time t, in seconds, since ou started the stopwatch. b. Assuming that d is a sinusoidal function of t, write a particular equation. Confirm b grapher that our equation gives the graph in part a. c. How high above or below the water s surface will point P be at time t 7.5 s? At that time, will P be going up or down? d. At what positive time t does point P first emerge from the water? a. From what s given, ou can tell the location of the sinusoidal ais, the high and low points, and the period. Sketch the sinusoidal ais at d 6 as shown in Figure 6-7c. Sketch the upper bound at d 6 7, or, and the lower bound at d 6 7, or. Sketch a high point at t. Because the waterwheel rotates at 6 rev/min, the period is 60, or 0 s. Mark the net high point at t 0, or. 6 Mark a low point halfwa between the two high points, and mark the points of inflection on the sinusoidal ais halfwa between each consecutive high and low. Sketch the graph through the critical points and the points of inflection. Figure 6-7c shows the finished sketch. b. d C A cos B(t D) From the graph, C 6 and A 7. D Horizontal dilation: 0 5 B 5 Additional Eploration Notes Eploration 6-7a can be used as a classroom group activit. It is significant because it requires students to find an -value far from the given piece of the graph. It also shows how a small change in initial Write the general equation. Use d and t for the variables. Cosine starts a ccle at a high point. The period of this sinusoid is 0; the period of the cosine function is. B is the reciprocal of the horizontal dilation. d 6 7 cos (t ) Write the particular equation. 5 Plotting on our grapher confirms that the equation is correct (Figure 6-7d). c. Set the window on our grapher to include 7.5. Then trace or scroll to this point (Figure 6-7d). From the graph, d , or 0.7 ft, and the graph is increasing, so point P is going up. conditions can make a large change in an etra p olated answer. Technolog Notes Eample : Waterwheel Problem asks several questions about a point traveling along a waterwheel that is partiall submerged in water. The eample is demonstrated in a Dnamic Precalculus Eploration at 0 Chapter 6: Applications of Trigonometric and Circular Functions

43 Problem Set 6-7 Quick Review Problems Q Q8 concern the circular function 4 5 cos ( 7). 6 Q. The amplitude is?. Q. The period is?. Q. The frequenc is?. Q4. The sinusoidal ais is at?. Q5. The phase displacement with respect to the parent cosine function is?. Q6. The upper bound is at?. Q7. If 9, then?. Q8. The first three positive -values at which low points occur are?,?, and?. Q9. Two values of arccos 0.5 are? and?. Q0. If 5, adding to the value of multiplies the value of b?. d. Point P is either submerging into or emerging from the water when d 0. At the first zero for positive t-values, shown in Figure 6-7d, the point is going into the water. At the net zero, the point is emerging. Using the intersect, zeros, or solver feature of our grapher, ou ll find that the point is at t s If ou go to ou can view the Waterwheel eploration for a dnamic view of the waterwheel and the graph of d as a function of t. Note that it is usuall easier to use the cosine function for these problems, because its graph starts a ccle at a high point. Reading Analsis. Steamboat Problem: Mark Twain sat on the deck of a river steamboat. As the paddle wheel turned, From what ou have read in this section, what do a point on the paddle blade moved so that its ou consider to be the main idea? What is the first distance, d, in feet, from the water s surface step in solving a sinusoidal model problem that was a sinusoidal function of time t, in seconds. takes it out of the real world and puts it into the When Twain s stopwatch read 4 s, the point was mathematical world? After ou have taken this step, at its highest, 6 ft above the water s surface. The how does our work in this chapter allow ou to wheel s diameter was 8 ft, and it completed a answer questions about the real-world situation? revolution ever 0 s. a. Sketch the graph of the sinusoid. 5min b. What is the lowest the point goes? Wh is it reasonable for this value to be negative? c. Find a particular equation for distance as a function of time. d. How far above the surface was the point when Mark s stopwatch read 7 s? e. 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 ou tell? f. Mark Twain is a pen name used b Samuel Clemens. What is the origin of that pen name? Give the source of our information. CAS Suggestions B this point in the chapter, students should have eplored and begun to master the techniques and implications of circular functions. Because the objective of this section is to appl the mathematics, not perform the algebraic manipulations, students benefit from using a CAS, which allows them to focus on what the mathematics mean. Defining the equation in part b of Eample and using that definition to graph and evaluate the model can help students solve part c. TI-Nspire users ma want to use a word or abbreviation to define the function. These figures show the definition of the waterwheel s height. The t-intercepts for part d can be found using either the Solve or the Zeros command. Section 6-7: Eploration 6-7: Chemotherap Problem can be done with sliders in Sketchpad or Fathom. Eploration 6-7a: Oil Well Problem in the Instructor s Resource Book can be done with sliders in Sketchpad or Fathom. CAS Activit 6-7a: Epicenter of an Earthquake in the Instructor s Resource Book has students eplore what information is required in order to find the epicenter of an earthquake. Students find the epicenter of an earthquake using triangulation. Allow 5 0 minutes. PROBLEM NOTES Supplementar Problems for this section are available at keonline. Q. 5 Q. Q. Q4. 4 Q5. 7 Q6. 9 Q Q8.,, 5 Q9. Q0. 9 See page 006 for answers to Problem. Section 6-7: Sinusoidal Functions as Mathematical Models

44 Problem Notes (continued) All of the problems in this section can be quite powerfull addressed b defining each function model on a CAS and, as appropriate, Using function notation to evaluate the models at specific points and create graphs without retping the function. Using the Solve command to determine independent variable values from given dependent values. Using domain restrictions after a Solve command to limit the number of responses to the desired values. Using the Zeros command instead of the Solve command when finding the independent variable s ais intercepts. Problem requires students to write an inequalit to answer part d. a., e. F b. F cos. (t.9) c. F (7) 7 foes; F (8) 8 foes; F (9) 77 foes; F (0) 77 foes d.. r t.5 r Problem describes onl half of a ccle. Students ma miss the period of the sinusoid if the draw or interpret the graph incorrectl. a. d b. d cos.5 (t 0.) c. d (7.) cm d. d (0) cm e. t s t t. Fo Population Problem: Naturalists find that populations of some kinds of predator animals var periodicall with time. Assume that the population of foes in a certain forest varies sinusoidall with time. Records started being kept at time t 0 r. A minimum number of 00 foes appeared at t.9 r. The net maimum, 800 foes, occurred at t 5. r. a. Sketch the graph of this sinusoid. b. Find a particular equation epressing the number of foes as a function of time. c. Predict the fo population when t 7, 8, 9, and 0 r. d. Suppose foes are declared a vulnerable species when their population drops below 00. Between what two nonnegative values of t did the foes first become vulnerable? e. Show on our graph in part a that our answer to part d is correct.. Bouncing Spring Problem: A weight attached to the end of a long spring is bouncing up and down (Figure 6-7e). As it bounces, its distance from the floor varies sinusoidall with time. You start a stopwatch. When the stopwatch reads 0. s, the weight first reaches a high point 60 cm above the floor. The net low point, 40 cm above the floor, occurs at.8 s. a. Sketch the graph of this sinusoidal function. b. Find a particular equation for distance from the floor as a function of time. c. What is the distance from the floor when the stopwatch reads 7. s? Chapter 6: Problem 4 involves a -variable that is a horizontal distance. Students ma find it easier to draw the graph if the turn their books so that the river in the illustration is up and the riverbank is down. 4a. 0 0 d. What was the distance from the floor when ou started the stopwatch? e. What is the first positive value of time when the weight is 59 cm above the floor? Floor 60 cm 40 cm Figure 6-7e 4. Rope Swing Problem: Zoe is at summer camp. One da she is swinging on a rope tied to a tree branch, going back and forth alternatel over land and water. Nathan starts a stopwatch. At time s, Zoe is at one end of her swing, at a distance ft from the riverbank (see Figure 6-7f). At time 5 s, she is at the other end of her swing, at a distance 7 ft from the riverbank. Assume that while she is swinging, varies sinusoidall with. 7 Riverbank Figure 6-7f River a. Sketch the graph of versus and write a particular equation. 5 0 cos ( ) 5 Chapter 6: Applications of Trigonometric and Circular Functions

45 b. Find when. s. Was Zoe over land or over water at this time? c. Find the first positive time when Zoe was directl over the riverbank ( 0). d. Zoe lets go of the rope and splashes into the water. What is the value of for the end of the rope when it comes to rest? What part of the mathematical model tells ou this? 5. Roller Coaster Problem: A theme park is building a portion of a roller coaster track in the shape of a sinusoid (Figure 6-7g). You have been hired to calculate the lengths of the horizontal and vertical support beams. a. The high and low points of the track are separated b 50 m horizontall and 0 m verticall. The low point is m below the ground. Let be the distance, in meters, a point on the track is above the ground. Let be the horizontal distance, in meters, a point on the track is from the high point. Find a particular equation for as a function of. b. The vertical support beams are spaced m apart, starting at the high point and ending just before the track goes below the ground. Make a table of values of the lengths of the beams. c. The horizontal beams are spaced m apart, starting at ground level and ending just below the high point. Make a table of values of horizontal beam lengths. Section 6-7: 4b ft; Zoe was over land. 4c s 4d. 5, the sinusoidal ais Problem 5 asks students to calculate the lengths of the horizontal and vertical support beams needed for a roller coaster. Some students double the horizontal beam lengths because the diagram shows onl half of a ccle. d. The builder must know how much support beam material to order. In the most timeefficient wa, find the total length of the vertical beams and the total length of the horizontal beams. Support beams Ground 50 m Figure 6-7g Track 0 m m 6. Buried Treasure Problem: Suppose ou seek a treasure that is buried in the side of a mountain. The mountain range has a sinusoidal vertical cross section (Figure 6-7h). The valle to the left is filled with water to a depth of 50 m, and the top of the mountain is 50 m above the water level. You set up an -ais at water level and a -ais 00 m to the right of the deepest part of the water. The top of the mountain is at distance 400 m. Mountaintop 50 Surface Water Treasure Figure 6-7h a. Find a particular equation epressing for points on the surface of the mountain as a function of. b. Show algebraicall that the sinusoid in part a contains the origin, (0, 0). c. The treasure is located beneath the surface at the point (0, 40), as shown in Figure 6-7h. Which would be the shorter wa to dig to the treasure, a horizontal tunnel or a vertical tunnel? Show our work. 5a. 5 5 cos 50 5b., 5c. See tables at right. 5d. Vertical timbers 4 m; horizontal timbers m Problem 6 shows onl half of a ccle. 6a cos 600 ( 400) 6b cos 600 (0 400) 5 0 m 6c. The vertical tunnel is shorter. 5b. 5c. Length 0 m 7 m m m 4 m m 6 m m 8 m m 0 m m m m 4 m m 6 m m 8 m m 0 m m m m 4 m m 6 m m 8 m m 0 m m m m 4 m m 6 m m 8 m m Length 0 m m m m 4 m m 6 m m 8 m m 0 m m m 5 m 4 m m 6 m m 8 m m 0 m m m m 4 m m 6 m m Section 6-7: Sinusoidal Functions as Mathematical Models

46 Problem Notes (continued) Problem 7 asks students to consult a reference in order to check the accurac of the model. One possible reference is the September 975 issue of Scientific American. 7a. r 7b. S cos (t 948) 7c. S(00) sunspots 7d. S (0) 7 sunspots; S (0) 5 sunspots; maimum in 05. 7e. The sunspot ccle resembles a sinusoid slightl but is not one. Problem 8 requires students to convert date and time information to the number of hours since midnight on August. 8a. d 5. 0.cos 5.5 (t 4) 8b.. m 8c. 6:0 a.m. 8d. 4:00:49 a.m. 8e. On the side closest to the Moon, the water is pulled more than Earth, causing a high tide. On the opposite side, farthest from the Moon, Earth is pulled more than the water, causing another high tide. Problem 9 is fairl straightforward and usuall does not present trouble for students. However, some students do not realize that at time 0 the vertical displacement of Earth is also zero; the ma start the graph at a minimum value instead of at zero. 7. Sunspot Problem: For several hundred ears, astronomers have kept track of the number of sunspots that occur on the surface of the Sun. The number of sunspots in a given ear varies periodicall, from a minimum of about 0 per ear to a maimum of about 0 per ear. Between 750 and 948, there were eactl 8 complete ccles. a. What is the period of a sunspot ccle? b. Assume that the number of sunspots per ear is a sinusoidal function of time and that a maimum occurred in 948. Find a particular equation epressing the number of sunspots per ear as a function of the ear. c. How man sunspots will there be in the ear 00? This ear? d. What is the first ear after 00 in which there will be about 5 sunspots? What is the first ear after 00 in which there will be a maimum number of sunspots? e. Find out how closel the sunspot ccle resembles a sinusoid b looking on the Internet or in another reference. 8. Tide Problem: Suppose ou are on the beach at Port Aransas, Teas, on August. At :00 p.m., at high tide, ou find that the depth of the water at the end of a pier is.5 m. At 7:0 p.m., at low tide, the depth of the water is. m. Assume that the depth varies sinusoidall with time. 4 Chapter 6: a. Find a particular equation epressing depth as a function of the time that has elapsed since :00 a.m. August. b. Use our mathematical model to predict the depth of the water at 5:00 p.m. on August. c. At what time does the first low tide occur on August? d. What is the earliest time on August that the water depth will be.7 m? e. A high tide occurs because the Moon is pulling the water awa from Earth slightl, making the water a bit deeper at a given point. How do ou eplain the fact that there are two high tides each da at most places on Earth? Provide the source of our information. 9. Shock Felt Round the World Problem: Suppose that one da all 00 million people in the United States climb up on tables. At time t 0, the all jump off. The resulting shock wave starts Earth vibrating at its fundamental period, 54 min. The surface first moves down from its normal position and then moves up an equal distance above its normal position (Figure 6-7i). Assume that the amplitude is 50 m. Jump! Down 50 m Figure 6-7i 50 m Up 50 m 50 m 4 Chapter 6: Applications of Trigonometric and Circular Functions