Post on 24-Aug-2020
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Unit6IntroductiontoTrigonometry
Lesson1:IncrediblyUsefulRatiosOpeningExerciseUserighttriangleΔABC toanswer1–3.
1. Namethesideofthetriangleopposite∠A
intwodifferentways.2. Namethesideofthetriangleopposite B∠
intwodifferentways.3. Namethesideofthetriangleopposite∠C
intwodifferentways.VocabularyWhenworkingwithanacuteanglewithinarighttriangle,weidentifythesidesofthetrianglebasedonitsrelationshiptothatmarkedangle.Thesideacrossfromtherightangle:Thesideacrossfromthemarkedangle:Thesidenexttothemarkedangle(notthehypotenuse):
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ExerciseForeachtriangle,labeltheappropriatesidesashypotenuse,opposite,andadjacentwithrespecttothemarkedacuteangle.
VocabularyThetrigonometricratiosarefunctionsofanangle,commonlyrepresentedwiththeGreekletterθ (pronounced“theta”).Theyrelatetheanglesofatriangletothelengthsofitssides.The3basictrig.ratiosaresine,cosineandtangentandhavethefollowingratios:sinθ =
cosθ =
tanθ =
Usingthetrianglepictured,identifythefollowing:
sinA= cosA= tanA=
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Examplea. Labelthesidesofeachtrianglewithrespecttothecircledangleashyp,opp,andadj.b. Findthefollowingtrig.ratios: sinA = sinB = cosA = cosB = tanA = tanB =
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Exercises1. Usingthetrianglepicturedbelow,identifythefollowingtrig.ratios:
a. sinA
b. cosA
c. tanA
d. sinB
e. cosB
f. tanB2. Usingthetrianglepicturedbelow,identifythefollowingtrig.ratios:
a. sinD
b. cosD
c. tanD
d. sinE
e. cosE
f. tanE
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Homework1. Usingthetrianglepicturedbelow,identifythefollowingtrig.ratios:
a. sinDb. cosDc. tanDd. sinEe. cosEf. tanE
2. Lookingatyouranswersfromquestion1,whatdoyounoticeaboutsinDandcosE?
WhataboutcosDandsinE?DoyouthinkthiswillhappenwithALLrighttriangles?Explain.
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Lesson2:SineandCosineofComplementaryAnglesOpeningExerciseUsingΔPQR ,completethefollowingtable:
θ sinθ cosθ tanθ
P
Q
Describeanypatternsyounoticeinthechart.Example1UsingΔABC ,completethefollowingtable(donotsimplifytheratios):
θ sinθ cosθ tanθ
A
B
DothepatternsfoundintheOpeningExercisestillwork?
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Example2Considertherighttriangle𝐴𝐵𝐶where∠𝐶isarightangle.Findthesumof∠ +∠A B .Justifyyouranswer.ImportantDiscovery!Theacuteanglesinarighttrianglearealwayscomplementary.Thesineofanyacuteangleisequaltothecosineofitscomplement.
90iffsin cosA B A B+ = = Usingthe2equationslistedabove,howcanwerewrite sinA = cosB intermsofoneangle?
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Exercises1. Findthevaluethatmakeseachstatementtrue:
a. sin cos32θ = ° b. sin 42 cos x° =
c. ( )cos sin 20θ θ= + ° d. ( )cos sin 30x x= − ° 2. InrighttriangleABCwiththerightangleatC, sinA = 2x + 0.1 and cosB = 4x − 0.7 . Determineandstatethevalueofx.Howcouldthisrelatebacktotherighttriangle?3. Explainwhy cos x = sin(90− x) forxsuchthat0 < x < 90 .
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Homework1. IfanglesPandQarethetwocomplementaryanglesofarighttriangle,
completethetable:
θ sinθ cosθ tanθ
P sinP = 11157
cosP = 6157
tanP = 116
Q
2. Findthevaluethatmakeseachstatementtrue:
a. cos12 sinθ= b. sin 2 cosx x= c. ( )sin cos 38x x= + ° d. ( )sin cos 3 10θ θ= + °
3. InrighttriangleABCwithrightangleatC,cosA= 4x + .07 and sinB = 2x + .13 . Determineandstatethevalueofx.Explainyouranswer.
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Lesson3:UsingTrigtoFindMissingSidesOpeningExercisea. Usingthegivendiagrams,findtheratiosforsinD andsin A .b. Reducetheseratios.Whatdoyounotice?c. Wouldthisalsoworkforcosineandtangent?Whyorwhynot?
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Wehavebeenworkingwithtrigonometricratiosforsine,cosine,andtangent.Wewillnowuseourcalculatorstofindthevaluesof θsin , θcos ,and θtan .YourgraphingcalculatormustbeinDEGREEMODEtobeabletocalculatethetrigvaluessincetheyarealldegreemeasures.DiscussionUnitCircle!Example1Useacalculatortofindthesineandcosineofθ .Roundyouranswertothenearestten-thousandth.
θ 0 10 20 30 40 50 60 70 80 90
θsin
θcos
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Example2Considerthegiventriangle.a. Usingtrigratios,findthelengthofsideatothenearesthundredth. b. Nowcalculatethelengthofsidebtothenearesthundredth.c. Whatmethod,otherthantrig,couldbeusedtodeterminethelengthofsideb?
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ExercisesFindthevalueofxtothenearesthundredth:1. 2.3. 4.
5. An8-footropeistiedfromthetopofapoletoastakeintheground,asshowninthediagrampictured.
Iftheropeformsa57°anglewiththeground,whatistheheightofthepole,tothenearesttenthofafoot?
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Homework In1-4,findthevalueofxtothenearesthundredth:
1. 2.3. 4.
5. Ahot-airballoonistiedtothegroundwithtwotaut(straight)ropes,asshowninthediagrambelow.Oneropeisdirectlyundertheballoonandmakesarightanglewiththeground.Theotherropeformsanangleof50° withtheground.
a. Determinetheheight,tothenearest
foot,oftheballoondirectlyabovetheground.
b. Determinethedistance,tothenearest
foot,onthegroundbetweenthetworopes.
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Lesson4:UsingTrigtoFindMissingAnglesOpeningExerciseThinkabouthowyousolvethefollowingequations:a. 2 14x = b. 2 9x =
c. Howdoyouthinkwewouldsolve 1sin2
x = ?
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Recallf
romyou
rtable
of
trigvaluestha
t
Intrigonometry,tosolve 1sin2
x = weneedtodotheinverseofsin ,whichisarcsin .
1 sin2
1arcsin(sin ) arcsin21arcsin(sin ) arcsin2
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x
x
x
x
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠
⎛ ⎞= ⎜ ⎟
⎝ ⎠=
Tosolve inyourcalculator:
o checkthatyourmodeisinDEGREES
o turntheequationinto
o whichisthesamethingas
o press andyourcalculatorwilldisplay
o typein as usingthedivisionkey
o hitentertoseetheanglemeasurethathasasinevalueof
Besuretoshowthisworkonyourpaper!
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Example1Findtheanglemeasurefromtheboytothetopofthetree.Roundyouranswertothenearesthundredth.
Example2Findthemeasureofthelabeledanglestothenearestdegree.
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Example4A16 footladderleansagainstawall.Thefootoftheladderis7 feetfromthewall.a. Findtheverticaldistancefromthegroundtothepointwherethetopoftheladder
touchesthewall.Roundyouranswertothenearesttenth.
b. Determinethemeasureoftheangleformedbytheladderandtheground.Roundyouranswertothenearestdegree.
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Exercises1. Findthemeasureofc tothenearestdegree. 2. Findthemeasureofd tothenearestdegree.3. Arollercoastertravels80ft oftrackfromtheloadingzonebeforereachingitspeak.
Thehorizontaldistancebetweentheloadingzoneandthebaseofthepeakis50ft .Atwhatangle,tothenearesttenthofadegree,istherollercoasterrising?
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Homework1. InrighttriangleABC, inches, inches,and .Findthe
numberofdegreesinthemeasureofangleBAC,tothenearestdegree.
2. Acommunicationscompanyisbuildinga30-footantennatocarrycellphonetransmissions.Asshowninthediagrambelow,a50-footwirefromthetopoftheantennatothegroundisusedtostabilizetheantenna.
Find,tothenearesthundredthofadegree,themeasureoftheanglethatthewiremakeswiththeground.
3. Asseenintheaccompanyingdiagram,apersoncantravel fromNewYorkCitytoBuffalobygoingnorth170milesto Albanyandthenwest280milestoBuffalo.
IfanengineerwantstodesignahighwaytoconnectNewYorkCitydirectlytoBuffalo,atwhatangle,x,wouldsheneedtobuildthehighway?Findtheangletothenearestdegree.
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Lesson5:AnglesofElevationandDepressionOpeningExerciseFromapoint120metersawayfromabuilding,Serenameasurestheanglebetweenthegroundandthetopofabuildingandfindsthatitmeasures °41 .Whatistheheightofthebuilding?Roundtothenearestmeter.
VocabularyAnotherwaytodescribethelocationofanglesinreal-worldrighttriangleproblemsistheangleofelevation(whenlookingupatanobject)andtheangleofdepression(whenlookingdownatanobject).Botharetheanglefoundbetweenthehorizontallineofsightandthesegmentconnectingthetwoobjects.Whatdoyouthinkistrueabouttherelationshipbetweentheangleofdepressionandtheangleofelevation?Explain.
x
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Example1 Amanstandingonlevelgroundis1000feetawayfromthebaseofa350-foot-tallbuilding.
Find,tothenearestdegree,themeasureoftheangleofelevationtothetopofthebuildingfromthepointonthegroundwherethemanisstanding.Example2Apersonmeasurestheangleofdepressionfromthetopofawalltoapointontheground.Thepointislocatedonlevelground62feetfromthebaseofthewallandtheangleofdepressionis52°.Tothenearesttenth,howfaristhepersonfromthepointontheground?Example3Scott,whoseeyelevelis1.5m abovetheground,stands30m fromatree.Theangleofelevationofabirdatthetopofthetreeis36° .Howfarabovethegroundisthebird?Roundyouranswertothenearesttenthofameter.
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Example4Samuelisatthetopofatowerandwillridedownaziplinetoalowertower.Thetotalverticaldropoftheziplineis40ft .Thezipline’sangleofelevationfromthelowertoweris11.5° .Tothenearesthundredth,howlongisthezipline?
Example5Amanwhois5feet8inchestallcastsashadowof8feet6inches.Assumingthatthemanisstandingperpendiculartotheground,whatistheangleofelevationfromtheendoftheshadowtothetopoftheman’shead,tothenearesttenthofadegree.
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Homework1. TheOccupationalSafetyandHealthAdministration(OSHA)providesstandardsfor
safetyattheworkplace.AladderisleanedagainstaverticalwallaccordingtoOSHAstandardsatanangleofelevationofapproximately75° .
a. Iftheladderis25ft.long,whatisthedistancefromthebaseoftheladderto
thebaseofthewall?Roundyouranswertothenearesttenth.
b. Tothenearesttenth,howhighonthewalldoestheladdermakecontact?
2. Standingonthegalleryofalighthouse(thedeckatthetopofalighthouse),apersonspotsashipatanangleofdepressionof20° .Thelighthouseis28mtallandsitsonacliff45mtallasmeasuredfromsealevel.Whatisthehorizontaldistancebetweenthelighthouseandtheship?Roundyouranswertothenearestwholemeter.
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Lesson6:TheLawofSinesOpeningExerciseGiventriangleDEF ,∠ = °22D ,∠ = °91F ,
=16.55DF ,and =6.74EF ,findDE tothenearesthundredth.WAIT!Canweanswerthisquestion?Whycan’tweusebasictrigorPythagoreanTheoremtofindDE?
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TheLawofSines
• Usedwhenworkingwith2sidesand2anglesofANYtriangle!(Notjustrighttriangles)
• Onlyuse2ofthefractions.
asinA
= bsinB
= csinC
Example1Let’strytheOpeningExerciseagainusingtheLawofSines!GiventriangleDEF ,∠ = °22D ,∠ = °91F ,
=16.55DF ,and =6.74EF ,findDE tothenearesthundredth.
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Exercises1. Intriangle ABC ,m 33A∠ = ° , 12a = ,andm 43B∠ = ° .Whatisthelengthofsidebto
thenearesthundredth?2. Inright PQRΔ ,m 90Q∠ = °,m 57P∠ = ° , 9.3p = .Findthemeasureofsideq,tothe
nearesttenth.3. In XYZΔ ,m∠Y =87° ,y=14andz=12.Findm∠Z ,tothenearesttenth.
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80ft
46°
28°
WaterTower
Ranger'sTower
C
B
A
Example2Afirebreaksoutonly80ftfromtheRanger’sTower.Therangerneedstoruntothewatertowertoopenthetowerspoutsothatitwillfloodtheareaandputoutthefire.Howfaristherunfromhistowertothewatertowertothenearestfoot?Example3Twolighthousesthatare30milesapartoneachsideoftheshorelinesrunnorthandsouth,asshown.Eachlighthousepersonspotsaboatinthedistance.Onelighthousenoticesthatthelocationoftheboatas 40° eastofsouthandtheotherlighthousemarkstheboatas 32° westofsouth.Whatisthedistancefromtheboattoeachofthelighthousesatthetimeitwasspotted?Roundyouranswertothenearestmile.
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Homework1. InΔBUG ,m∠B =78° ,m∠G =42° ,and g=99 .Findthelengthofsideb,tothe
nearesttenth.2. In XYZΔ ,m 125Y∠ = ° ,m 18X∠ = ° ,!!y =112 .Findthelengthofsidez,tothe
nearesthundredth. 3. Abaseballfanissittingdirectlybehindhomeplateinthelastrowoftheupperdeck.
Theangleofdepressiontohomeplateis30° andtheangleofdepressiontothepitcher’smoundis 24° .Thedistancebetweenthepitcher’smoundandhomeplateis60.5feet.Howfar,tothenearestfoot,isthefanfromhomeplate?
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Lesson7:Trig.ApplicationsOpeningExerciseAnarchaeologicalteamisexcavatingartifactsfromasunkenmerchantvesselontheoceanfloor.Toassisttheteam,aroboticprobeisusedremotely.Theprobetravelsapproximately3,900metersatanangleofdepressionof67.4degreesfromtheteam’sshipontheoceansurfacedowntothesunkenvesselontheoceanfloor.Thefigureshowsarepresentationoftheteam’sshipandtheprobe.Howmanymetersbelowthesurfaceoftheoceanwilltheprobebewhenitreachestheoceanfloor?Giveyouranswertothenearesthundredmeters.
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Example1Asshownbelow,acanoeisapproachingalighthouseonthecoastlineofalake.Thefrontofthecanoeis1.5feetabovethewaterandanobserverinthelighthouseis112feetabovethewater.
At5:00,theobserverinthelighthousemeasuredtheangleofdepressiontothefrontofthecanoetobe 6° .Fiveminuteslater,theobservermeasuredandsawtheangleofdepressiontothefrontofthecanoehadincreasedby 49° .Determineandstate,tothenearestfootperminute,theaveragespeedatwhichthecanoetraveledtowardthelighthouse.
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Example2ThemapbelowshowsthethreetallestmountainpeaksinNewYorkState:MountMarcy,AlgonquinPeak,andMountHaystack.MountHaystack,theshortestpeak,is4960feettall.SurveyorshavedeterminedthehorizontaldistancebetweenMountHaystackandMountMarcyis6336feetandthehorizontaldistancebetweenMountMarcyandAlgonquinPeakis20,493feet.
TheangleofdepressionfromthepeakofMountMarcytothepeakofMountHaystackis3.47degrees.TheangleofelevationfromthepeakofAlgonquinPeaktothepeakofMountMarcyis0.64degrees.Whataretheheights,tothenearestfoot,ofMountMarcyandAlgonquinPeak?Justifyyouranswer.
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Homework1. Timisdesigningarooftrussintheshapeofanisoscelestriangle.Thedesignshows
thebaseanglesofthetrusstohavemeasuresof18.5° .Ifthehorizontalbaseoftherooftrussis36ft.across,whatistheapproximatelengthofonesideoftheroof?Roundyouranswertothenearesthundredth.
2. Aradiotowerisanchoredbylongcablescalledguywires
shownas AB and AD inthegivendiagram.Point A is250m fromthebaseofthetower.IftheangleofelevationfrompointAtopointDis 71° andtheangleofelevationtopointBis65° ,findtothenearestmeterthedistancebetweenpointsBandD.
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Lesson8:SpecialRightTrianglesOpeningExerciseTherearecertainspecialangleswhereitispossibletogivetheexactvalueofsineandcosine.Thesefrequentlyseenanglesare °0 , °30 , °45 , °60 ,and °90 .
Usingthegiventriangles,completethefollowingtableandrationalizethedenominatorsifnecessary.
𝜽 𝟎˚ 𝟑𝟎˚ 𝟒𝟓˚ 𝟔𝟎˚ 𝟗𝟎˚
Sine 0 𝟏
Cosine 1 𝟎
Findtwovaluesinthetablethatarethesame.Whatdoyounoticeabouttheanglemeasures?Findadifferentsetofvaluesinthetablethatarethesame.Whatdoyounoticeabouttheiranglemeasures?
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RatioofSidesofSpecialRightTriangles30–60–90triangle 45–45–90triangle
2 : 2 3 : 4 2 : 2 : 2 2 3 : : 3 : :
4 : : 4 : :
: : x : : x Example1Findtheexactvalueofthemissingsidelengthsinthegiventriangle.Example2Findtheexactvalueofthemissingsidelengthsinthegiventriangle.
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Example3Findtheexactvalueofthemissingsidelengthsinthegiventriangle.ExercisesIn1-4,findtheexactvalueofthemissingsidesusingspecialrighttriangles.1. 2.3. 4.
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HomeworkIn1-4,findtheexactvalueofthemissingsidesusingspecialrighttriangles.1. 2. 3. 4.5. Givenanequilateraltrianglewithsidesoflength9,findthelengthofthealtitude
usingspecialrighttriangles.ConfirmyouranswerwiththePythagoreanTheorem.
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Lesson9:TrigonometryandthePythagoreanTheoremOpeningExercise
Inarighttrianglewithanacuteangleofmeasureθ , 1sin2
θ = .
a. Drawthetriangleandlabeltheangleandknownsidelengths.b. Findtheexactlengthofthemissingsideofthetriangle.c. Findtheexactvalueforcosθ .d. Findtheexactvaluefor tanθ .
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Example1
Inarighttrianglewithanacuteangleofmeasureθ , 7sin9
θ = .Usingthesameprocessas
theOpeningExercise,whatisthevalueof tanθ insimplestradicalform?Example2Inlesson3wediscoveredthat (x, y) = (cosθ, sinθ ) .Usingthisinformation,wearegoingtodiscovertwomoretrigidentities! QuotientIdentity PythagoreanIdentity
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Example3WearegoingtoredotheproblemsfromtheOpeningExerciseandExample1usingthePythagoreanIdentity!FromtheOpeningExercise:
Inarighttriangle,withacuteangleofmeasureθ , 1sin2
θ = .UsethePythagoreanIdentity
todeterminetheexactvalueofcosθ andthenusetheQuotientIdentitytofind tanθ .FromExample1:
Inarighttriangle,withacuteangleofmeasureθ , 7sin9
θ = .UsethePythagoreanIdentity
todeterminetheexactvalueofcosθ andthenusetheQuotientIdentitytofind tanθ .
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Homework
1. If 4cos5
θ = ,findsinθ and tanθ .
2. If 5sin5
θ = ,findcosθ and tanθ .
3. If tan 5θ = ,findsinθ andcosθ .