mrs. bisgaard's class - home - name: group members ......pre-calculus chapter 6 notes section...

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


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?


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)



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.




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


Theperiod:

Thehorizontaltranslation(forcotangent):



y

36° θ 21°9° 6° 51°

1

4. WriteaparticularequationforthegraphinProblem3.Checkyouranswerbyplottingonyourgrapher.

5. ForyH1C3csc4(θD10−),give


Theperiod:

Thehorizontaltranslation:



6. SketchthegraphofyH1C3csc4(θD10−),showingverticalasymptotes,horizontalaxis,andcriticalpoints.

y

θ

7. Forthenextgraph,give


Theperiod:

Thehorizontaltranslation(forsecant):



y

θ

70°20° 160°

1

4

7

8. WriteaparticularequationforthegraphinProblem7.Checkyouranswerbyplottingonyourgrapher.



Trigonometric Graphs

𝒚 = 𝐬𝐢𝐧 𝜽 𝒚 = 𝐜𝐬𝐜 𝜽

𝒚 = 𝐜𝐨𝐬 𝜽 𝒚 = 𝐬𝐞𝐜 𝜽

𝒚 = 𝐭𝐚𝐧 𝜽 𝒚 = 𝐜𝐨𝐭 𝜽


Radian Lab


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) =



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



Section 6.5 Notes

Trigonometric Functions

Inputs:

Inverses:

All 3 Inverse Trigonometric Functions


Circular Functions

Inputs:

Inverses:



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


y

x55 10 15 20 25

16

2



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?


y

x55 10 15 20 25

16

2


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



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


ChemotherapyProblem:Awomanhascancerandmusthaveachemotherapytreatmentonceevery3weeks.Onesideeffectisthatherredbloodcellcountgoesdownandthencomesbackupbetweentreatments.OnJanuary13(day13oftheyear),shegetsatreatment.Atthattime,herredbloodcellcountisatahighof800.Halfwaybetweentreatments,thecountdropstoalowof200.Assumethattheredbloodcellcountvariessinusoidallywiththedayoftheyear,x.

y (red cell count)

10 20 30 40 50

x (days)

1000



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.


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

mrs. bisgaard's class - home - name: group members ......pre-calculus chapter 6 notes section...

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