I understand the new topics for this unit if I can do the practice questions in the textbook/handouts

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1 1 U n i t 6 11U Date: Name: Sinusidals Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will learn hw trignmetry can be used t mdel wavelike relatinships. These wavelike functins are called sinusidal. Yu will study key prperties that these functins have and use these prperties t sketch these functins, t mdel real life situatins and t slve trignmetric equatins. In grade 12 yu will cntinue studying these functins but instead f degree mde yu will learn hw t use the radian mde. Crrectins fr the textbk answers: Sec 6.1 #2b) perid = 3 #4e) perid = 5 Sec 6.3 #6 all shuld start at MIN #8b) 20cm c) 362cm Sec 6.6 #13 y=-30cs(1.9x) withut runding, if yu rund yu can get very different answers!! #14 y=7cs(22.7x)+8 Sec 6.7 #5a) -30cs(18(t-2)) #5b) -9cs(18(t-2)) #9 a) k=1.4 Success Criteria I am ready fr this unit if I am cnfident in the fllwing review tpics Simplifying expressins Slving equatins Factring SOH CAH TOA Sine and Csine laws Transfrmatins Functin ntatin Inverses Dmain & range Radicals Unit circle Exact trig ratis fr special angles Trig identities I understand the new tpics fr this unit if I can d the practice questins in the textbk/handuts Date pages Tpics # f quest. dne? Yu may be asked t shw them Peridic Functin Prperties Sectin 6.1 Sinusidal Functin Prperties Sectin 6.2 Interpreting Sinusidals Sectin 6.3 Transfrmatins & Sketching Sinusidals Sectin 6.4 & 6.5 Mdelling with Sinusidals Sectin 6.6 & Handut Slve Prblems with Sinusidals Sectin 6.7 & Handut REVIEW Reflect previus TEST mark, Overall mark nw. Lking back, what can yu imprve upn? 1

2 2 U n i t 6 11U Date: Name: Peridic Functin Prperties There are many situatins in real life that repeat in cycles.fr example, tides, daylight hurs, temperature fr the year, heartbeat, vlume f air in lungs, rides n ferris wheels, the list can g n. This trend that repeats in cycles is called peridic phenmenn. The length f the cycle is called the perid. The average value f peaks and trughs is the axis f the functin, and the distance frm the axis t the maximum, r frm the axis t the minimum is called the amplitude. 1. Summarize the equatins that yu can use t find the perid frm the graph, the axis, the amplitude and the range. 2. After the sun rises, its angle f elevatin increases rapidly at first, then mre slwly, reaching a maximum in 26 weeks. Then the angle decreases until sunset. a. Wuld yu cnsider this trend peridic? Explain. b. When des sunrise ccur at this time f the year, fr this particular spt n Earth? c. What is the perid? What des it represent? What is lnger the night r the day fr this situatin? d. What is the axis? What des it represent? e. What is the amplitude? What des it represent? f. What is the range? What des it represent? g. Extraplate the angle f elevatin in 32 weeks and interplate the angle f elevatin in 12 weeks. h. During what weeks is the angle f elevatin f the sun abve 30 degrees? 2

3 3 U n i t 6 11U Date: Name: 3. The mvement f a factry machine that cuts grves in metal t create a required template pattern is shwn n the fllwing graph. y x a. Describe what the machine culd be ding at each part f the graph. Wuld yu cnsider this trend peridic? b. What is the axis, amplitude, range and perid? c. Hw fast is the machine mving n its way up? 4. Decide if each graph r descriptin r table f values is peridic r nt. a. b. c. d. e. Dependent = the hrizntal distance travelled by the grandfather clck s pendulum Independent = time f. Dependent = interest n the mney invested at 5% Independent = principal depsited g. Dependent = the height f the pedal n a mving bicycle Independent = minutes h. x y i. x y Fr all the graphs and tables f peridic situatins state the values f the perid, axis, amplitude, and range. 3

4 4 U n i t 6 11U Date: Name: 6. Sketch a peridic graph with the fllwing characteristics: a. perid f 8 units, amplitude f 3, and axis at 2 b. perid r ¾ units and range between 4 and 1 7. A buy, bbbing up and dwn in the water f a depth f 16 feet. As waves mve past it, the buy mves frm its highest pint t its lwest pint and back t its highest pint. The water n average has 5 waves every 40 secnds. The distance between its highest and lwest pints is 3 feet. a. Sketch fr 3 cycles and state the dmain. b. State the increasing and decreasing intervals. 4

5 5 U n i t 6 11U Date: Name: Sinusidal Functin Prperties Sme f the questins in the textbk require yu t graph with technlgy. There are lts f applets yu can use nline, r yu can dwnlad a free prgram t use n yur cmputer ffline. Online Graphing Calculatr Dwnlad GeGebra (ffline and nline) select webstart, fr ffline select appletstart, fr nline 1. Use technlgy t graph the fllwing and decide if they are peridic r nt. Sketch small pictures belw. a. b. c. d. e. x y = x 2 y = (1.1) sin 2x y = 10sin x 15 tan Graphing technlgy ften des nt have degrees as independent variable. It has radians which yu will learn in gr.12 Fr nw, if the equatin has a degree symbl in it set k π 180 = as a multiple f the given k value s that the graph will appear crrectly. y = 4sin 2x cs x y = 3cs(2 x 4) 2. Sinusidal graphs resemble a regular symmetrical lking wave. Only tw f the abve peridic functins are cnsidered sinusidal. Can yu guess which nes? 3. It is time t develp the parent graphs f the functins that give a sinusidal wave. Place and label the x and y axes in the fllwing graphs in the prper places s that it wuld represent a sine functin and a csine functin. Sine graph Csine graph 4. Fr each functin, utline ne cmplete cycle and highlight the 5 key pints n the cycle. State reminders f hw t begin sketching each type f graph. 5

6 6 U n i t 6 11U Date: Name: 5. Recall the meaning f sine and csine ratis. Cnnect these meanings t why sine starts at the axis and csine starts at max. 6. In general any pint n the circle can be defined as a. Fr unit circles, f radius 1: b. Fr circles f radius r: c. What is the new lcatin f the pint (6, 0) that was rtated abut 55? 7. Fr f ( x) = sin x and g( x) cs x = sketch bth graphs n the same grid Find all the x values in the dmain 180 x 180 such that a. f ( x ) = 0 b. g( x ) = 1 c. f ( x) = g( x) 6

7 7 U n i t 6 11U Date: Name: 8. All twers and skyscrapers are designed t sway with the wind. When standing n the glass flr f the CN twer the equatin f the hrizntal sway is h( x) = 40sin( x), where h is the hrizntal sway in centimetres and x is the time in secnds. a. Use table f values t help yu graph this: x h b. What is the perid? What des it represent? c. State the maximum and minimum values f sway and the times at which they ccur. d. State the mean value f sway and the time at which it ccurs. e. If a guest arrives n the glass flr at time = 0, hw many secnds will have elapsed befre the guest has swayed 20 cm frm the hrizntal? USING GRAPH: USING EQUATION f. Find h (2.034), what des it represent? 7

8 8 U n i t 6 11U Date: Name: Interpreting Sinusidals 1. Tw weights attached t the end f tw springs are buncing up and dwn. As they bunce their height, in cm, varies with time, in sec, as shwn in the graph: Cmpare and cntrast the bunces f these weights. 8

9 9 U n i t 6 11U Date: Name: 2. The ppulatin, F, f fxes in the regin is mdelled by the functin F( t) = 500sin(15 t) , where t is the time in mnths. The ppulatin, R, f rabbits in the same regin is mdelled by the functin, R( t) = 5000sin(15 t + 90) a. Graph F(t) and R(t). Use technlgy t help yu. Graphing technlgy ften des nt have degrees as independent variable. It has radians which yu will learn in gr.12 Fr nw, if yur equatin has a degree symbl in it set k π 180 = as a multiple f the given k value value s that the graph will appear crrectly. b. State the maximum and minimum values and the mnth in which they ccur fr bth species in the chart belw Mnth fr Mnth fr Mnth fr Max Value Min Value Mean Value Max Min Mean Fx Rabbit c. Describe the relatinships between the maximum, minimum and mean pints f the tw curves in terms f the lifestyles f the rabbits and fxes and list pssible causes fr the relatinships. 9

10 10 U n i t 6 11U Date: Name: 3. As yu ride a Ferris wheel, yur distance frm the grund varies sinusidally with time accrding t the equatin h( t) = 14sin(5( t 24)) + 16 where h is height in meters and t is time in secnds. The graph f this mdel is belw a. What is the radius f the wheel? What part f the equatin gives yu this? b. Where is the centre f the wheel lcated? What part f the equatin gives yu this? c. Hw lng des it take fr this Ferris wheel t cmplete ne full revlutin? What part f the equatin gives yu this? d. If the Ferris wheel was sped up, what part f the equatin will change? e. If the Ferris wheel rtated in the ther directin, sketch the resulting graph n the same grid. What part f the equatin will change? f. Hw far ff the grund did yu bard the Ferris wheel? g. Yu tk a vide f the whle ride. The vide is 7 minutes 12 secnds lng, hw many revlutins did the wheel make during this ride? h. At 3 minutes and 14 sec f the vide yu were level with a nearby building, hw tall is that building? i. When was the last time yu were at maximum height befre yu had t get ff? 10

11 11 U n i t 6 11U Date: Name: Transfrmatins & Sketching Sinusidals 1. In real life it is very rare t have degrees as the independent variable. Usually it is time r hrizntal distance. What type f transfrmatin can cancel the degrees ut? 2. Yu can als tell frm the equatin whether degrees is the unit t be used n the x-axis. Lk at the placement f the degree symbl in the equatin and decide whether x represents degrees r if the degrees are cancelled ut. a. y sin3 = x b. y = sin( x 7 ) c. y = sin x + 5 d. y = sin(2x 4) + 8 Nte: In grade 12, yu will learn hw t use radian mde and yu will n lnger have t write the degree symbl, since radian measure actually has n units. Therefre, equatins with n units, like y = sin x + 5, are assumed t be in radians unless within the questin it says the dmain is 0 x Review what kind f transfrmatins d the cnstants cntrl in the equatins y = acs( k( x d)) + c r y = asin( k( x d)) + c. 4. State the transfrmatins f the fllwing. a. y = 2cs(0.75 x) + 3 b. y = 2.5sin(2x 60 ) 5. State the dmain and range fr the functins abve if yu are tld yu want t have 3 cycles in the dmain. 11

12 12 U n i t 6 11U Date: Name: Yu can sketch sinusidals in 3 ways: by applying step-by-step transfrmatins t the parent functin using different clur fr each step OR by using a table f values OR by using key characteristics 6. PRACTICE sketching using different methds a. y = 2cs(0.75 x) step by step methd using different clurs fr each transfrmatin, applied in crrect rder b. y = 2.5sin(2x 60 ) - table f values methd 12

13 13 U n i t 6 11U Date: Name: 7. Summarize hw t sketch by finding all the key characteristics frm a given equatin. 8. State all the key characteristics f the fllwing then sketch a. y = 2sin(3θ + 180) 4 b. y = 3cs(2x 120)

14 14 U n i t 6 11U Date: Name: c. y = 50 cs(2θ 270) 10 d. y = 235sin(8x + 288) Find the equatin fr csine graph if yu are tld that the range is 6 y 2 and perid is 240. Assume there are n reflectins r hrizntal shifts. 14

15 15 U n i t 6 11U Date: Name: Mdelling with Sinusidals 1. Summarize hw yu can find the equatin frm a given graph. 2. Find tw pssible equatins fr sine and tw pssible equatins fr csine fr each f the fllwing a. b. 3. In the graphs abve, ntice that x-axis is labelled differently, hw des that affect the equatin that yu write dwn? 15

16 16 U n i t 6 11U Date: Name: 4. Find ne pssible equatin fr sine and ne fr csine fr each f the fllwing a. mre cmplicated example b. 5. Find an equatin fr the fllwing graph 16

17 17 U n i t 6 11U Date: Name: 6. The average mnthly temperature f a city in degrees Fahrenheit are given belw Jan Feb Mar Apr May June July Aug Sept Oct Nv Dec a. Sketch the data b. Find an apprximate equatin that wuld mdel this data. Assume that x=1 is the start f January. c. Use the equatin t find the apprximate mnthly temperature fr the middle f August. d. Mst f the husehlds turn n the heat if the temperature falls belw 64 F. Fr what dmain d mst husehlds use heating in their hmes? e. What part(s) f the equatin will change if the data was taken frm a warmer climate? 17

18 18 U n i t 6 11U Date: Name: 7. Summarize hw t find the perid and the k value if yu are given the speed in revlutins per secnd. Use the example, 10 revlutins in 35 secnds t help yu explain. 8. A cnveyr belt is pwered by tw circular pulleys, ne f radius 5m the ther f 3m. Bth pulleys have serial numbers n them, but when the cnveyr belt starts, the serial numbers are in different psitins. The bigger pulley, A, has the serial number starting at maximum height, the smaller pulley, B, has the serial number starting at minimum height which is 2m away frm the flr. The pulleys are als spinning at different speeds but have the same minimum height. Pulley A cmpletes 5 revlutins in 2 minutes. Pulley B is faster, cmpleting 5 revlutins in a minute. a. Find the perid fr each pulley. b. Sketch bth functins. c. Find the equatins that mdel the heights. d. Analyze the meaning f all cnstants and variables fr the equatin that describes pulley A, in the cntext f the prblem. 18

19 19 U n i t 6 11U Date: Name: Slve Prblems with Sinusidals 1. A weight attached t the end f a lng spring is buncing up and dwn. As it bunces, its distance frm the flr varies sinusidally with time. Yu start a stpwatch. When the stpwatch reads 0.3 secnds, the weight first reaches a high pint 60 cm abve the flr. The next lw pint, 40 cm abve the flr, ccurs at 1.8 secnds. (Dn t assume that at time zer the weight is at the minimum!) a. Sketch a graph f this sinusidal functin. b. Write an equatin expressing distance frm the flr in terms f the number f secnds the stpwatch reads. c. What was the distance frm the flr when yu started the watch? d. Predict the time at which the weight was 59 cm abve the flr fr the first time and fr the secnd time. 19

20 20 U n i t 6 11U Date: Name: 2. Yu are n an 8-seat Ferris wheel at an unknwn height when the ride starts. It takes yu 24 secnds t reach the tp f the wheel 26m abve the grund. The lading platfrm is 4m high. The wheel makes a cmplete revlutin in 60 secnds. a. Create a diagram/sketch t shw yur lcatin n the ride and the key features f the wheel. b. Find the sinusidal equatin that mdels yur ride n the Ferris wheel. c. Hw high abve the grund will yur seat be after 90 secnds? d. Find the tw times within ne cycle when the height is at 24m. HINT fr TIPS p372 #8 text uses distance travelled. Use circumference frmula C = 2 r fr hrizntal distance travelled by the wheel. π 20

21 21 U n i t 6 11U Date: Name: 3. Yu are at Risser s Beach, N.S., t search fr interesting shells. At 2:00 p.m. n June 19, the tide is in (i.e., the water is at its deepest). At that time yu find that the depth f the water at the end f the breakwater is 15 meters. At 8:00 p.m. the same day when the tide is ut, yu find that the depth f the water is 11 meters. Assume that the depth f the water varies sinusidally with time. a. Derive an equatin expressing depth in terms f the number f hurs that have elapsed since 12:00 nn n June 19. b. Use yur mathematical mdel t predict the depth f the water at 7:00 a.m. n June 20. c. At what time will the first lw tide ccur n June 20? d. What is the earliest time n June 20 that the water will be at 12.7 meters deep? 21

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec

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