mrs. bisgaard's class - home - name: group members ......pre-calculus chapter 6 notes section...
TRANSCRIPT
132 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-1a: Sine and Cosine Graphs, Manually Date:
Objective: Find the shape of sine and cosine graphs by plotting them on graph paper.
3. Findsin45−andcos65−.ShowthatthecorrespondingpointsareonthegraphsinProblems1and2,respectively.
4. FindtheinversetrigonometricfunctionsθHsinD10.4andθHcosD10.8.ShowthatthecorrespondingpointsareonthegraphsinProblems1and2,respectively.
5. Whataretherangesofthesineandcosinefunctions?
6. Nameareal-worldsituationwherevariablesarerelatedbyaperiodicgraphlikesineorcosine.
7. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
1. Onyourgrapher,makeatableofvaluesofyHsinθforeach10−from0−to90−.Setthemodetoroundto2decimalplaces.Plotthevaluesonthisgraphpaper.AlsoplotyHsinθforeach90−through720−.Connectthepointswithasmoothcurve,observingtheshapeyouplottedfor0−to90−.
y
90° 180° 270° 360° 450° 540° 630° 720°
1
1
θ
2. PlotthegraphofyHcosθpointwise,thewayyoudidforsineinProblem1.
y
90° 180° 270° 360° 450° 540° 630° 720°
1
1
θ
Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 133© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-2: Periodic Daily Date:
Temperatures alternate version
Objective: Apply a transformed sinusoid to model the average daily high temperature at a particular location as a function of time.
Month Temperature(−F) Month Temperature(−F)
July 94.9
Aug. 94.6
Sept. 89.3
Oct. 81.5
Nov. 70.7
Dec. 64.6
Jan. 61.7
Feb. 66.3
Mar. 73.7
Apr. 80.3
May 85.6
June 91.8
1. Onthegraphpaper,plottheaveragedailyhightemperaturesfortwoyears.AssumethatJanuaryismonth1andsoforth.Determineatime-efficientwayforyourgroupmemberstodotheplotting.Whatshouldyouplotformonthzero?Connectthepointswithasmoothcurve.
y
x6 12 18 24
100
90
80
70
60
50
40
30
20
10
Months
Tem
per
atu
re (°F
)
2. ThegraphofyHcosθcompletesacycleeach360−(angle,nottemperature).Whathorizontaldilationfactorwouldmakeitcompleteacycleeach12−,asshown?Writeanequationforthistransformedsinusoidandplotitonyourgrapher.
y
θ
12° 24°
1
1
3. Earthrotates360−aroundtheSunin12months.HowdothesenumbersrelatetothedilationfactoryouusedinProblem2?
4. ThetemperaturegraphinProblem1hasahighpointatxH7months.WhattransformationwouldyouapplytothesinusoidinProblem2(dashedinthenextfigure)tomakeithaveahighpointatθH7−(solid)insteadofatθH0−?Writetheequationandconfirmitbyplottingitonyourgrapher.
y
θ
12°
7°
24°
1
1
5. Theaverageofthehighestandlowesttemperaturesinthetableis94.9C61.7________
2H78.3.Writeanequationfor
thetransformationthatwouldtranslatethegraphinProblem4upwardby78.3units.
6. The94.9highpointinProblem1is16.6unitsabove78.3,andthe61.7lowpointis16.6unitsbelow78.3.WriteanequationforthetransformationthatwoulddilatethesinusoidinProblem5byafactorof16.6sothatitlookslikethisgraph.Confirmyouranswerbygrapher.
y
θ
7°
61.778.3
94.9
7. Onyourgrapher,plotthepointsyouplottedinProblem1.HowwelldoesthesinusoidalequationinProblem6fitthepoints?
8. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
HereareaveragedailyhightemperaturesforSanAntonio,bymonth,basedondatacollectedoverthepast100yearsandpublishedbyNOAA,theNationalOceanicandAtmosphericAdministration.Suchdataareused,forexample,inthedesignofheatingandairconditioningsystems.
Pre-Calculus Chapter 6 Notes
Section 6.2 Notes
|A| is the __________________________ (A is the _____________________________________,
which can be positive or negative.)
B is the ____________________ of the ________________________ dilation.
C is the location of the _______________________________________ (vertical translation).
D is the _____________________________________________ (horizontal translation).
Examples:
Write the equation of the sinusoid
using cosine & sine.
Sketch the sinusoid on the graph below.
𝒚 = 𝟓 + 𝟒𝐬𝐢𝐧𝟐(𝛉 − 𝟑𝟎°)
General Sinusoidal Equations
𝒚 = 𝑪 + 𝑨 𝐜𝐨𝐬 𝐁(𝜽 − 𝑫) 𝒚 = 𝑪 + 𝑨 𝐬𝐢𝐧 𝐁(𝜽 − 𝑫)
136 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-3a: Tangent and Secant Graphs Date:
Objective: Discover what the tangent and secant function graphs look like and how they relate to sine and cosine.NographersallowedforProblems1–7.
1. Thereciprocalpropertystatesthat
secθH 1______ cosθ
Withoutyourgrapher,usethispropertytosketchthegraphofyHsecθonthesameaxesasthegraphoftheparentfunctionyHcosθ.Inparticular,showwhathappenstothesecantgraphwherevercosθH0.
y
540°450°360°270°180°90°90°
1
θ
2. Writethequotientpropertyexpressingtanθasaquotientoftwoothertrigonometricfunctions.
3. ThenextfigureshowstheparentfunctionsyHsinθ andyHcosθ.BasedonyouranswertoProblem2,determinewheretheasymptotesareforthegraphofyHtanθ,andmarkthemonthefigure.
y
540°450°360°270°180°90°90°
1
θ
4. Basedonthequotientproperty,findoutwheretheθ-interceptsareforthegraphofyHtanθ.MarktheseinterceptsonthefigureinProblem3.
5. AtθH45−,sinθ andcosθareequal.Basedonthisfact,whatdoestan45−equal?MarkthispointonthegraphinProblem3.MarkallotherpointswheresinθHcosθ.
tan45−H
6. UsethepointsandasymptotesyouhavemarkedtosketchthegraphofyHtanθonthefigureinProblem3.(Nographersallowed!)
7. Checkyourgraphswithyourinstructor.
Graphersallowedfortheremainingproblems.
8. Onyourgrapher,plotthegraphofyHcscθ.Sketchtheresulthere.
9. Onyourgrapher,plotthegraphofyHcotθ.Sketchtheresulthere.
10. AtwhatvaluesofθarethepointsofinflectionforyHtanθ?Explainwhythetangentfunctionhasnocriticalpoints.
11. ExplainwhythegraphofyHsecθhasnopointsofinflection,eventhoughthegraphgoesfromconcaveuptoconcavedownatvariousplaces.
12. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 137© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-3b: Transformed Tangent Date: and Secant GraphsObjective: Sketch transformed tangent, cotangent, secant, and cosecant graphs, and find equations from given graphs.
1. ForyH3C1__ 2tan5(θD7−),state
Thehorizontaldilation:
Theperiod:
Thehorizontaltranslation:
Theverticaldilation:
Theverticaltranslation:
2. SketchthegraphofyH3C1__ 2tan5(θD7−),showing
verticalasymptotes,horizontalaxis,pointsofinflection,andothersignificantpoints.
y
θ
3. Forthenextgraph,state
Thehorizontaldilation:
Theperiod:
Thehorizontaltranslation(forcotangent):
Theverticaldilation:
Theverticaltranslation:
y
36° θ 21°9° 6° 51°
1
4. WriteaparticularequationforthegraphinProblem3.Checkyouranswerbyplottingonyourgrapher.
5. ForyH1C3csc4(θD10−),give
Thehorizontaldilation:
Theperiod:
Thehorizontaltranslation:
Theverticaldilation:
Theverticaltranslation:
6. SketchthegraphofyH1C3csc4(θD10−),showingverticalasymptotes,horizontalaxis,andcriticalpoints.
y
θ
7. Forthenextgraph,give
Thehorizontaldilation:
Theperiod:
Thehorizontaltranslation(forsecant):
Theverticaldilation:
Theverticaltranslation:
y
θ
70°20° 160°
1
4
7
8. WriteaparticularequationforthegraphinProblem7.Checkyouranswerbyplottingonyourgrapher.
9. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
Pre-Calculus Chapter 6 Notes
Trigonometric Graphs
𝒚 = 𝐬𝐢𝐧 𝜽 𝒚 = 𝐜𝐬𝐜 𝜽
𝒚 = 𝐜𝐨𝐬 𝜽 𝒚 = 𝐬𝐞𝐜 𝜽
𝒚 = 𝐭𝐚𝐧 𝜽 𝒚 = 𝐜𝐨𝐭 𝜽
Pre-Calculus Chapter 6 Notes
Radian Lab
Pre-Calculus Chapter 6 Notes
Section 6.4 Notes
Radian-Degree Conversion
To find the radian measure of 𝜃, multiply the degree measure by _________.
To find the degree measure of 𝜃, multiply the radian measure by _________.
Examples:
Find the exact radian measure.
135°
Find the approximate radian measure.
34°
Find the exact degree measure.
5𝜋
6
Find the approximate degree measure.
0.33 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Examples:
Find the exact values.
sin (2𝜋
3) = csc (
5𝜋
4) = sec(4𝜋) =
Find the approximate values.
cos(4) = sec(2) = cot−1(4) =
140 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-5a: Circular Function Parent Graphs Date:
Objective: Plot circular function sinusoids and tangent graphs.
1. SketchtheparenttrigonometricfunctionyHsinθ.
y
720°360°
1
1
θ
2. SketchtheparenttrigonometricfunctionyHcosθ.
y
720°360°
1
1
θ
3. SketchtheparenttrigonometricfunctionyHtanθ.
720°360°
y
1
1
θ
4. Setyourgraphertoradianmode.Setthewindowwith0K x K4πandthey-valuesasshownonthegivengraphs.ThenplotthegraphofthecircularfunctionyHsinx.Sketchtheresult.
y
2π 3π 4ππ
1
1
x
5. Withyourgrapherstillinradianmode,plotthegraphofthecircularfunctionyHcosx.Sketchtheresult.
y
2π 3π 4ππ
1
1
x
6. Withyourgrapherstillinradianmode,plotthegraphofthecircularfunctionyHtanx.Sketchtheresult.
4π3ππ 2π
y
1
1
x
7. Theonlydifferencebetweentheparentgraphsforthecircularfunctionsinusoidandtheordinarytrigonometricfunctionsinusoidistheperiod.Explainhowtheperiodsofthetwotypesofsinusoidrelatetodegreesandradians.
8. Thegraphhereisatransformedcircularfunctionsinusoid.Usingwhatyouhavelearnedabouttransformations,findaparticularequationofthissinusoid.Confirmbygrapherthatyourequationiscorrect.
y
x10
1
9. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
Pre-Calculus Chapter 6 Notes
Section 6.5 Notes
Trigonometric Functions
Inputs:
Inverses:
All 3 Inverse Trigonometric Functions
Pre-Calculus Chapter 6 Notes
Circular Functions
Inputs:
Inverses:
Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 141© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-6a: Sinusoids, Given y, Date: Find x NumericallyObjective: Find a particular equation for a given sinusoid and use it to graphically and numerically find x-values for a given y-value.
1. Forthesinusoidshown,drawthelineyH5.Readfromthegraphthesixvaluesofxforwhichthelinecrossesthepartofthegraphshown.Writeyouranswerstoonedecimalplace.
xH , , ,
, , .
2. Writeanequationforthissinusoid.
3. PlottheequationfromProblem2onyourgrapher.Doesitlooklikethegivengraph?
4. TraceyourgraphinProblem3toxH17.Doesyourgraphhaveahighpointthere?
5. CircletheleftmostpointonthegivengraphatwhichyH5.PlotthelineyH5,andusetheintersectfeaturetofindthevalueofxatthispoint.
xH
6. OthervaluesofxforwhichyH5canbefoundbyaddingmultiplesoftheperiodtothevalueofxinProblem5.Letnbethenumberofperiodsyouadd.FindtwomorevaluesofxforwhichyH5.Circlethethreex-valuesinProblem1thatarealsoanswerstoProblem5andthisproblem.
Multiple,nH1: xH
Multiple,nH2: xH
7. Putaboxonthefigureatapointwhosex-valueisnotananswertoProblem5or6.Usetheintersectfeaturetofindoneofthesex-values.
xH
8. Addmultiplesoftheperiodtothex-valuesinProblem5or7tofindtheothertwox-valuesthatarealsoonthegraph.Tellwhatmultipleoftheperiodyouadded.
Multiple,nH : xH
Multiple,nH : xH
9. ByaddinganappropriatemultipleoftheperiodtotheanswertoProblem5or7,findthefirstvalueofxgreaterthan1000forwhichyH5.Atthisvalueofx,willybeincreasingordecreasing?Howcanyoutell?
Multiple,nH : xH
10. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
y
x55 10 15 20 25
16
2
142 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-6b: Given y, Find x Algebraically Date:
Objective: Given the particular equation for a sinusoid and a value of y, calculate the corresponding x-values algebraically.
1. Thesinusoidhastheequation
yH9C7cos2π ___ 13(xD4)
ConfirmthatthisequationgivesthecorrectvalueofywhenxH15.
2. YourobjectiveistofindalgebraicallythevaluesofxgivenyH5.Substitute5fory.Thendothealgebranecessarytogetxusinganarccosine.Writethegeneralsolutionintheform
xH(number)C(period)nor(number)C(period)n
3. WritethetwovaluesofxfromthegeneralsolutioninthenH0rowofthistable.Byaddingandsubtractingmultiplesoftheperiod,fillintheotherrowsinthetablewithmorepossiblevaluesofx.
n x1 x2
D1
0
1
2
4. CirclethepointsonthegivengraphwherethelineyH5cutsthegraph.Foreachpoint,tellthevalueofnatthatpoint.
5. FindthetwovaluesofxifnH100.
6. Findthefirstvalueofxgreaterthan1000forwhichyH5.Whatdoesnequalthere?
7. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
y
x55 10 15 20 25
16
2
Pre-Calculus Chapter 6 Notes
Section 6.7 Notes
Examples:
Find the first five positive values of the
inverse circular function.
arccos 0.9
Solve the equation for the 5 values of x
shown on the graph below.
1 − 3 cos𝜋
8(𝑥 − 1) = 1.5
Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 143© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-7: Chemotherapy Problem alternate version Date:
Objective: Use sinusoids to predict events in the real world.
1. Drawthegraphofthesinusoidonthegivenaxes.Showenoughcyclestofillthegraphpaper.
2. Writeaparticularequationforthe(circular)sinusoidinProblem1.Itisrecommendedthatyouusethecosinefunction.
3. Enteryourequationintoyourgrapher.Plotthegraphusingthewindowshown.Explainhowthegraphverifiesthatyourequationiscorrect.
4. Thewomanfeels“good”iftheredbloodcellcountis700ormore,“bad”ifthecountis300orless,and“so-so”ifthecountisbetween300and700.Howwillshebefeelingonherbirthday,March19?Explainhowyouarrivedatyouranswer.
5. Showonyourgraphtheintervalofdatesbetweenwhichthewomanwillfeel“good”asshecomesbackfromthelowpointaftertheJanuary13treatment.
6. FindpreciselythevaluesofxatthebeginningandendoftheintervalinProblem5bysettingyH700andusingappropriatenumericorgraphicalmethods.Describewhatyoudid.
xH andxH
7. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
ChemotherapyProblem:Awomanhascancerandmusthaveachemotherapytreatmentonceevery3weeks.Onesideeffectisthatherredbloodcellcountgoesdownandthencomesbackupbetweentreatments.OnJanuary13(day13oftheyear),shegetsatreatment.Atthattime,herredbloodcellcountisatahighof800.Halfwaybetweentreatments,thecountdropstoalowof200.Assumethattheredbloodcellcountvariessinusoidallywiththedayoftheyear,x.
y (red cell count)
10 20 30 40 50
x (days)
1000
144 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press
Name: Group Members:
Exploration 6-7a: Oil Well Problem Date:
Objective: Use sinusoids to predict events in the real world.
1. Findaparticularequationforyasafunctionofx.
2. Plotthegraphonyourgrapher.UseawindowwithD100K x K900.Describehowthegraphconfirmsthatyourequationiscorrect.
3. Findgraphicallythefirstintervalofx-valuesintheavailablelandforwhichthetopsurfaceoftheformationisnomorethan1600feetdeep.Drawasketchshowingwhatyoudid.
4. FindalgebraicallythevaluesofxattheendsoftheintervalinProblem3.
5. SupposethattheoriginalmeasurementswereslightlyinaccurateandthatthevalueofxshownatD65feetwasatxH D64instead.WouldthisfactmakemuchdifferenceintheanswertoProblem3?Useatime-efficientmethodtoreachyouranswer.Explainwhatyoudid.
6. Whatdidyoulearnasaresultofdoingthisexplorationthatyoudidnotknowbefore?
Thefigureshowsaverticalcrosssectionthroughapieceofland.They-axisisdrawncomingoutofthegroundatthefenceborderinglandownedbyyourboss,EarlWells.Earlownsthelandtotheleftofthefenceandisinterestedinbuyinglandontheothersidetodrillanewoilwell.Geologistshavefoundanoil-bearingformation,whichtheybelievetobesinusoidalinshape,beneathEarl’sland.AtxH D100feet,thetopsurfaceoftheformationis,atitsdeepest,yH D2500feet.Aquarter-cycleclosertothefence,atxH D65feet,thetopsurfaceisonly2000feetdeep.Thefirst700feetoflandbeyondthefenceisinaccessible.EarlwantstodrillatthefirstconvenientsitebeyondxH700ft.
y
y = 2500 ft
100 65 30
Top surface
Fence
y = 2000 ft
x = 700 ft
xInaccessible land Available land