UNIT FOUR TRIGONOMETRIC FUNCTIONS MATH 621B 25 HOURS

Transcription

1 UNIT FOUR TRIGONOMETRIC FUNCTIONS MATH 621B 25 HOURS Revised April 9, 02 73

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3 Trigonometric Function Introductory Lesson C32 create scatter plots of periodic data and analyse using appropriate data Student groups could investigate data that will be periodic. An example is shown below. After student groups do the example they could be asked to think of other examples that would be periodic. An instance could be graph the population of the Island over two full years. They could estimate that the population increases greatly in the summer and declines back in the fall and finally back to normal in the winter. These are problems where the students get to think about quantities in the real world that cycle in time. Example: From the website, following data was obtained for average monthly temperatures at Charlottetown over a year. If a student so chooses another community on PEI or anywhere for that matter could have been chosen. a) Draw a graph (use the TI-83 if desired) of the data b) Explain the pattern in the data c) Why is the graph called a periodic graph? d) Predict the shape of the graph over the next year and extend the above graph to include the second year. e) What do you think is the period of the graph(time to repeat the pattern)? f) Sketch a graph of the temperature for a full year if you lived in Hawaii? Average monthly maximum temperatures in C on PEI Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec!3.4! !0.3 75

4 SCO: By the end of grade 12, students will be expected to: D16 understand the connection between degree and radian measure and be able to use both Elaborations - Instructional Strategies/Suggestions Angular Measure (4.1) Invite student groups to read and discuss p The differences between degree measure and radian measure should be discussed. Students should be able to convert from one unit of angular measure to the other. One radian is the measure of the angle formed by rotating the radius of a circle through an arc length equal to the radius. Challenge the students to do the Explore and Inquire on p.186. Angles written without a degree symbol are assumed to be in radian measure. Students should become familiar with terms such as: < radius < arc length < sector of a circle < sector angle Students will be expected to draw angles in standard position and become acquainted with terms such as: < Initial arm < Terminal arm < direction of rotation ( + and! angles) < co-terminal angles < principal angle 76

5 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Angular Measure (4.1) Pencil/Paper Draw each of the following angles in standard position. Label the diagram completely and correctly including the direction of rotation. a) 100 d) 765 b)!250 e) 2 π 3 Angular Measure (4.1) Math Power 12 p.189 #1,2,3-19 odd 21-28,29-39 odd 41-46,47-57 odd, 50 Applications p.190 # 71,73,74,77,81,83 3 c) f) 4 π 7π 3 Pencil/Paper Convert from one angle measure to the other: a) 30 d) 5 π 4 π b) e)!75 2 c) 150 f) 2 π 5 Journal What is the size of 1 radian in degrees? Write to explain how you determined it? Communication Explain to the other members of your group what a sector angle is. Presentation Explain how different arc lengths can have the same size sector angle. 77

6 SCO: By the end of grade 12, students will be expected to: C52 extend sine and cosine ratios and functions to all angle measures and apply them Elaborations - Instructional Strategies/Suggestions Exact Trig Ratios (4.2) In scientific work it is sometimes advantageous to work with exact trig ratios. Invite student groups to do the Explore and Inquire on p.194. Students should then be introduced to the 3 reciprocal trig functions as defined on the top of p.195. Allow time for the groups to get the values of the 3 reciprocal trig ratios for the Explore diagram on p.194. Allow time for student groups to read and discuss Ex.1 on p.195 and come to an agreement on the meaning of reference angle and generate the diagrams below for standard angle 2 and reference angle ": B11 derive, analyse and apply angle and arclength relationships using the unit circle Invite student groups to complete the table on p.196 which illustrates the signs of the trig functions for all 4 quadrants. One way to help students remember is the phrase: ASTC: All Students Take Chemistry indicates the positive primary trig ratio(s) in each quadrant. Note: Template of unit circle at the end of the unit. 78

7 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Exact Trig Ratios (4.2) Pencil/Paper Draw the angle in standard position if its terminal arm contains the point (!2,!3). Label the reference angle. Find the exact values of the 6 trig ratios. Group Activity If sec 2 =!5/8, and 2 is in standard position having its terminal arm in the second quadrant, find the exact values of the other 5 trig ratios. Exact Trig Ratios (4.2) Math Power 12 p.199 #1-33 odd,35-42 Applications #44,45,49 Activity Write the exact value of: a) sin 300 b) csc! 7B/6 Group Activity Complete the following table using exact values: 79

8 SCO: By the end of grade 12, students will be expected to: C39 analyse tables and graphs of sine and cosine equations to find patterns C49 translate between graphical and algebraic representations of sinusoidal models Elaborations - Instructional Strategies/Suggestions Graphing Trig Functions (4.3) Student groups should do the : Graphing sin 2 and cos 2 Investigation at the end of the unit Graphing sin x and cos x Investigation at the end of the unit Student groups should read the investigation Period and Amplitude on p.202 and do the related exercises. Students should be able to come to a consensus on definitions of period and amplitude. An important concept for students to appreciate is the Periodicity of the trig functions and the Amplitude or height of the wave. The general sine function can be written as: y = a sin b( x + c) + d In this section we will investigate the effect of the coefficients a and b on the basic y = sin x function. Challenge students to do the: a Investigation for the sin function at the end of the unit b Investigation for the sin function at the end of the unit They should be able to discover that : period P = 2π b amplitude a = max 2 min 80

9 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Graphing Trig Functions (4.3) Pencil/Paper What is the amplitude of each function? a) y = 4cosθ b) y = 3 4 sin θ c) y = 5sinθ Pencil/Paper Find the period, in degrees, of each function: a) y = cos 3θ b) y = sin 8θ c) y = 4 sin 2θ Graphing Trig Functions (4.3) Do the Graphing sin 2 and cos 2 Investigation at the end of the unit. Do the Graphing sin x and cos x Investigation at the end of the unit. Do Period and Amplitude Activity p Do the a Investigation for the sin function at the end of the unit. Do the b Investigation for the sin function at the end of the unit. Pencil/Paper Write an equation for the sine function defined by: a) amplitude 3, period 90 b) amplitude 2, period B/4 c) Math Power 12 p.209 #1-39 odd, Applications p.210 #55,57, Project Find the birth date of Katrina Lemay Doan, then find her biorhythm index on the day she won the speed skating Olympic gold medal in

10 SCO: By the end of grade 12, students will be expected to: C49 translate between graphical and algebraic representations of sinusoidal models Elaborations - Instructional Strategies/Suggestions Phase Shift and Horizontal Translation (4.4) Student groups should do the c and d Investigation at the end of the unit as an introduction to the section. Students will now investigate the c and d coefficients in the formula: y = a sin b( x c) + d Invite student groups to do the Explore and Inquire on p.212. For the sine or cosine functions the coefficients have the following effect: a = vertical stretch or amplitude b = horizontal stretch or period c = horizontal translation d = vertical translation Student groups should examine and discuss examples 1-5 on p Students should appreciate that the variable after the sine or cosine function does not have to represent angle measure but in fact can be many other real world quantities. In order for it to represent degrees the degree symbol must be present otherwise the quantity is understood to be in radians. 82

11 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Phase Shift and Horizontal Translation (4.4) Group Activity Determine the amplitude, period, phase shift and vertical displacement for: a) y = 3sin x + 1 π b) y = 2 cos 2( x + ) π c) y = 2 sin ( x ) Phase Shift and Horizontal Translation (4.4) Do c and d Investigation at the end of the unit. Do a, b, c and d Investigation at the end of the unit Math Power 12 p.218 #1-31 odd,32 Applications p.219 #33-35,37 Activity From the graph below determine the equation if: a) the function is a sine function b) the function is a cosine function Possible solutions: y = π 2 sin 2( x ) 2 y = 2 sin 2( x + π ) π y = 2 cos 2( x + ) 4 3π y = 2 cos 2( x ) 4 Journal Write to explain what effect the coefficients c and d have on the basic sine or cosine function. Group Project Do the Investigating Sound Waves activity on p (Subject to availability of probes) 83

12 SCO: By the end of grade 12, students will be expected to: C2 describe real world relationships depicted by graphs, tables of values and written descriptions C31 model situations with sinusoidal curves Elaborations - Instructional Strategies/Suggestions Applications (4.5) This section offers opportunities for students to investigate many of the applications of sine and cosine functions. Student groups should do the Explore and Inquire on p.222. Challenge students to read and discuss the examples of sinusoidal applications on p Note to Teachers: This is a great chance for students to research projects for assessment purposes. Student groups should do the Applications Worksheet and the Harmonic Motion in a Spring activities at the end of the unit. C38 determine the equation for the curve of best fit using sinusoidal regression available on technology 84

13 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Applications (4.5) Group Project Contact at least 10 businesses and/or professionals and conduct an interview about what mathematics is used in their profession. Applications (4.5) Do the Applications Worksheets at the end of the unit. Math Power 12 p.225 #1-11 Group Activity On the internet find data that exhibits periodic behaviour. Use the TI-83 to model the data and determine the equation that defines the data. Research Project Use the internet or a resource such as a Physics, Chemistry, Biology, Economics or Sociology text to illustrate 3 examples of where a sine or cosine function defines a real world situation. Write to explain these examples. Applications p.226 #16,17,21,22 Applications of Trigonometry video (Math Dept in each High School has a copy) Experiment Perform the experiment on Harmonic Motion. See the end of the unit for the lab. 85

14 SCO: By the end of grade 12, students will be expected to: B40 explore and analyse the graphs of the reciprocal trigonometric functions Elaborations - Instructional Strategies/Suggestions Other Trig Functions (4.6) Invite students to read and discuss the other trig functions as explained on p Initially, challenge student groups to do the Explore and Inquire on sin θ p Students will see that tan θ = cos θ and there are asymptotes where the denominator equals 0, ( ). 2 + nπ Students should do the tan investigation worksheet at the end of the unit. Similar to the general sine and cosine functions these general functions look much the same: y = a tan b( x c) + d y = a cot b( x c) + d y = a csc b( x c) + d y = a sec b( x c) + d π Where a, b, c and d have the same effect as on the sine or cosine functions. Students should understand that these other four trig functions are periodic but they have restrictions on their domains resulting in vertical asymptotes. On the TI-83 graph the sine function and the csc function. Examine the tables of values to determine that the function values actually are reciprocals of each other. This can be repeated for the other two pairs of functions as well. Ex. Students should do the csc, sec and cot investigation at the end of the unit. 86

15 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Other Trig Functions (4.6) Technology Graph the following on the TI-83 using window [!B/2,B/2,B/4] by [!3.3,1]: a) y = tan x b) y = 2tanx c) y = 1 x 2 tan Other Trig Functions (4.6) Math Power 12 p.232 #1-15 Applications p.232 #16,18,19 Worksheets at end of unit. d) What effect does a have on the basic tangent function? Technology Graph the following on the TI-83 using window [!B/2,B/2, B/4] by [!3.3,1]: a) y = tan x b) y = tan 2x c) y = tan 1 x 2 d) What effect does b have on the basic tangent function? Technology Graph sin x and csc x on the same TI-83 screen (use zoom 7: trig). Then press 2 nd table and contrast the two tables of values for these functions. How are the corresponding y values related to each other? Is this reflected in the graphs? Unit Circle (4.2) 87

16 Unit Circle (4.2) 88

17 Graphing sin 2 and cos 2 Investigation (4.3) Use your calculator to complete the following table(round to 3 places of decimals) and neatly graph the data on the TI-83 screens shown below sin2 cos2 Each on on the horizontal scale represents 30. divisi a) Graph y = sin 2 using the TI-83. Use the window dimensions indicated below. Press 2 nd Tblset then press 2 nd Table. Compare this table with the one above. Are they the same? b) Repeat part (a) with y = cos 2 and compare the tables of values. c) What is the period of the sine function in degrees? Of the cosine function? d) What maximum and minimum values do the sine and cosine functions cycle between? e) What is the amplitude of each of the functions? 89

18 Graphing sin x and cos x Investigation(4.3) Complete the following table (round to 3 place of decimals) and neatly graph the data on the TI-83 screens shown below. 2 0 B/6 B/3 B/2 2B/3 5B/6 B 7B/6 4B/3 3B/2 5B/ 3 11B/6 2B sin x cos x Each division on the horizontal scale represents B/6 radians. a) Graph y = sin x using the TI-83. Use the window indicated below. Then press 2 nd Tblset and press 2 nd Table. Compare this table with the one above. Are they the same? b) Repeat part (a) with y = cos x and compare the tables of values. c) What is the period of the sine function in radians? Of the cosine function? d) What maximum and minimum values do the sine and cosine functions cycle between? e) What is the amplitude of each of the functions? 90

19 a Investigation for the sin function(4.3) Explore the role of a using the TI-83. Graph all the following on the same screen. Then sketch approximate graphs on the TI screens below: a) y = sin x b) y = 2sinx c) y = 3sin x d) y = sin x e) y = 2sinx f) y = 3sin x g) What effect does a have on the basic sine function? 91

20 b Investigation for the sin function(4.3) Explore the role of b using the TI-83. Graph all the following on the TI-83. Then sketch approximate graphs on the TI screens below: a) y = sin x b) y = sin 2x c) y = sin 3x d) What effect does b have on the basic sine function? c Investigation for the sin function(4.4) 92

21 Explore the role of c using the TI-83. Graph all the following on the same screen using the same dimensions as in the a investigation, then sketch approximate graphs on the graph paper below: a) y = sin x π b) y = sin( x + ) 4 π c) y = sin( x ) 4 d) y = sin x π e) y = sin( x + ) 2 π f) y = sin( x ) 2 f) What effect does c have on the basic function? 93

22 d Investigation for the sin function(4.4) Explore the role of d using the TI-83. Graph all the following on the same screen. Then sketch approximate graphs on the graph paper below: a) y = sin x b) y = sin x + 1 d) y = sin x + 3 e) y = sin x f) y = sin x 1 e) y = sin x 3 f) What effect does d have on the basic function? a, b, c and d Investigation(4.4) 94

23 Each function below has the form, y = a sin( x c) + d or y = a cos( x c) + d. Write the equation for each graph with the given window dimensions and mode shown below. Check your answer using the calculator

24 a) a sine function: b) a cosine function:

25 Temperatures July 7, 1995 at Charlottetown(4.5) The temperatures in the table on the right were taken by Environment Canada at Charlottetown on July 7, On graph paper plot the points and draw a graph for the data shown. Let t = the number of hours since midnight. Using the table at the right find the: Amplitude Period Vertical translation Determine the phase shift for the graph. Write an equation for a sine function that will match the graph. Enter the time (as hours since midnight) in L 1 and the temperatures in L 2. Press STAT PLOT Press enter on 1:Plot 1 Set your calculator to look like the screen on the right To draw the graph for this data press and then select 9:ZoomStat ZOOM Press Y= calculator is set in radians. and enter your equation. Make sure the Does you graph match the data given? If not adjust your equation to have it fit the data. The new equation is 97

26 Time Temperature 12:00 am :00 am :00 am :00 am :00 am :00 am :00 am :00 am :00 am 19.1 Amplitude Period Vertical Translation Equation for the Sine Function This does not quite fit right. We must account for the phase shift (horizontal translation). Estimate the phase shift and write this new equation. 9:00 am 21 10:00 am :00 am :00 pm :00 pm :00 pm 26.1 How does this fit the data? You may have to re-adjust the phase shift c coefficient a number of times. Your equation of best fit is 3:00 pm :00 pm :00 pm 25 6:00 pm :00 pm :00 pm :00 pm :00 pm :00 pm 19 12:00 am 18 98

27 Applications Worksheet(4.5) Time Height (m) Time Temp ( C) 12 am am am am am am am am am am am am pm pm pm pm pm pm 25 6 pm pm pm pm pm pm am am am am am am am am am am am am pm pm pm pm pm pm pm pm pm pm pm pm am am 18 The temperatures shown above were taken at Charlottetown over a 24 hour period. Enter the data into L 1 and L 2 in the TI-83. Plot the data and determine 99

28 The data shown above is a record of the heights of the local tides over a 24 hour period. Enter the data into L 1 and L 2 in the TI-83. Plot the data and determine: Amplitude Period Vertical Translation Equation for the Cosine Function This does not quite fit right. We must account for the phase shift (horizontal translation). Estimate the phase shift and write this new equation. How does this fit the data? You may have to re-adjust the phase shift c coefficient a number of times. Your equation of best fit is 100

29 Divisions are B/4 The graph is y = 2 sin x 101

30 Divisions are B/4 The graph is y = 2 cos x 102

31 Tangent Investigation (4.6) 1. Graph y = tan x on the TI-83 using the zoom 7: trig window. a) What is the period of this function? b) Examine the table of values ( 2 nd table) and write to explain the existence of asymptotes. 2. Graph the following on the TI-83 using window [!B/2, B/2, B/4] by [!3.3,1]: a) y = tan x b) y = 2tanx c) y = 1 x 2 tan d) What effect does a have on the basic tangent function? 3. Graph the following on the TI-83 using window [!B/2,B/2, B/4] by [!3.3,1]: a) y = tan x b) y = tan 2x c) y = tan 1 x 2 d) What effect does b have on the basic tangent function? 4. Graph the following on the TI-83 using window [!B, B, B/4] by [4, 4, 1]: a) y = tan x π b) y = tan( x + ) 4 π c) y = tan( x ) 4 d) What effect does c have on the basic function? 5. Graph the following on the TI-83 using window [!B/2,B/2, B/4] by [!4.4,1]: a) y = tan x b) y = tan x + 2 c) y = tan x 2 d) What effect does d have on the basic function? 103

32 Reciprocal Investigation (4.6) 1. Graph y = tan x and y = cot x on the same TI-83 screen (use zoom 7: trig). Then press 2 nd table and contrast the two tables of values for these functions. a) How are the corresponding y values related to each other? b) Is this reflected in the graphs? c) What is the period of the cot function? d) Predict the effect of the coefficients a, b, c, and d on the basic cotangent function? Discuss your predictions with the other members in your group and come to a consensus on their effects. 2. Graph y = sin x and y = csc x on the same TI-83 screen (use zoom 7: trig ). Then press 2 nd table and contrast the two tables of values for these functions. a) How are the corresponding y values related to each other? b) Is this reflected in the graphs? c) What is the period of the cot function? d) Predict the effect of the coefficients a, b, c, and d on the basic co-secant function? Discuss your predictions with the other members in your group and come to a consensus on their effects 3. Graph y = cos x and y = sec x on the same TI-83 screen (use zoom 7: trig ). Then press 2 nd table and contrast the two tables of values for these functions. a) How are the corresponding y values related to each other? b) Is this reflected in the graphs? c) What is the period of the cot function? d) Predict the effect of the coefficients a, b, c, and d on the basic secant function? Discuss your predictions with the other members in your group and come to a consensus on their effects 104

33 Harmonic Motion in a Spring OBJECT: To determine the period and amplitude of a spring. APPARATUS: Slinky, pendulum clamp, stand, 50 g mass hanger, slotted 100 g mass, Ranger, TI- 83, metre stick. PROCEDURE: Work in pairs. Set up the apparatus as shown in the diagram above. Directly below the mass place the Ranger. Stretch the mass 15 cm below the equilibrium position. Have a partner record the time required for 7 cycles while the second partner audibly counts the cycles. Record the data in the table below and calculate the period for one cycle. Amount stretched T(7 cycles) T( 1 cycle) Use of the Ranger: Press pgrm choose Ranger and press enter twice. Choose 1:Setup/sample and use the following settings: Real Time No Time(s) 5 Display Dist Begin on Enter Smoothing None Units Metres Cursor up to the top of the screen and over to Start Now and press enter. When you are ready to start collecting data, press enter again and the data will be collected and the graph displayed. Once the graph is displayed use trace to determine the period from peak to peak. Period = 105

34 max min Use trace to get the maximum and minimum to determine the amplitude a = = 2 How do the period and amplitude agree with your experimental observations? To exit the Ranger graph, press enter 6:Quit The data collected is also stored in L 1 and L 2 in the TI-83. Do a statplot of this data. Press 2 nd statplot 1:Plot 1 turn On and choose the second type of graph and press zoom :9 zoomstat Then use sinusoidal regression to get the equation of the function. Press stat < calc? SinReg press enter 2 nd L 1, 2 nd L 2, vars < Yvars 1:Function 1:Y 1 then press enter to begin the regression. The regression equation is : Reading the equation, determine the amplitude a Does it agree with the amount that you stretched the spring? What period is indicated in the regression equation? Does it agree with the experimental period found in the table? Write a brief summary of your conclusions from this experiment. 106

35 Use of Ranger A Check to what programs are in your TI-83. Prgm, if Ranger is not displayed then B to enter the Ranger program 1) connect the Ranger to the TI-83 using the cable provided. 2) 2 nd Link < enter 3) open the pivoting head on the Ranger(CBR) and press 82/83 calculator should display RECEIVING. When the transfer is complete, the green light on the Ranger flashes once, the CBR beeps once and the calculator displays DONE 4) disconnect the cable 5) press 2 nd quit Transferring the Ranger Program to other calculators Use the short cords that came with the TI-83 to connect the calculator with the Ranger Program to on that doesn t have the program. 2 nd Link 3:Pgrm? Ranger then press < 1:Transmit (enter) For the second TI-83 that will be receiving the program: 2 nd Link < RECEIVE enter and it will say done when the program has been transferred. Check to see if the Ranger Program is in the second calculator. 107

36 Waves on the TI-83 The superposition principle states that if two or more waves exist simultaneously in the same region of space the displacement of the disturbance is the linear combination (algebraic sum) of the individual waves. A periodic wave can be represented by: Look at 3 cases from Physics: F HG y = A sin 2π x + φ λ I K J A = amplitude 8 = wavelength N = phase 1) Sawtooth wave y = 1 sin( nx) On the TI-83 show the first 5 terms. Set the Mode n to Degree. n = 1 2) Square wave y = 1 sin( nx) using the same dimensions n n= odds L N M n 112 / ( 1) odds 3) Triangular wave y = sin( nx) using the same dimensions 2 n O Q P 108

37 RMS current For direct current the maximum value, the average value and the instantaneous value of the current are all the same. The power dissipated in heat in a resistor by direct current is: P = 2 I R For ac current, the average value over a whole cycle is zero but ac current still produces heat in 2πt ( i = Imax sin 2πυt = Imax sin ) where L(frequency) = 1/T ( T = period) T a resistor. If the instantaneous value of the current is i, then i 2 is always positive, even though i is negative for one-half of each cycle. For ac current, the power dissipated in a resistor at any instant is 2 i 2 P = i 2 R and the average power dissipated over each cycle is i R where is the average value of the square of the current. The effective value of an ac current, I eff, is the value of that direct current which would produce heat at the same rate as the ac current in a given resistor. The effective value of an ac current is found by taking the average of the square of the instantaneous current and then extracting its square root. Hence the name root-mean-square (rms) current. I max The effective value of a sine-wave current is Ieff = 2 The effective value of an ac potential difference (voltage) is V V eff = max Ordinarily ac ammeters and voltmeters are calibrated to read effective values. When we speak of a 110 volt ac current we are talking about effective values. Derivation Let the ac current be i t = I max sin 2π T The power dissipated instantaneously in a resistor R is I R sin 2 t T 2 2 2π max 109

38 The power dissipated during one cycle is: T eff sin zmax 0 I RT = I R πt T dt zt πt Therefore I T I T dt I eff = max sin = max T I 2 eff 2 I max = 2 I max Ieff = 2 Also V V eff = max 2 110

39 Trig Rule Template Trig Rule Template 111

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