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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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28

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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 .