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Section 5: Introduction to Trigonometry and Graphs The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC §111.42(c). 5.01 Radians and Degree Measurements Precalculus (4)(B) Precalculus (4)(C) Precalculus (4)(D) 5.02 Linear and Angular Velocity Precalculus (4)(D) 5.03 Trigonometric Ratios Precalculus (4)(E) 5.04 Trigonometric Angles and the Unit Circle Precalculus (2)(P) Precalculus (4)(A) 5.05 Graphs of Sine and Cosine Precalculus (2)(F) Precalculus (2)(G) Precalculus (2)(I) Precalculus (2)(O) 5.06 Graphs of Secant and Cosecant Precalculus (2)(F) Precalculus (2)(G) Precalculus (2)(I) Precalculus (2)(O) 5.07 Graphs of Tangent and Cotangent Precalculus (2)(F) Precalculus (2)(G) Precalculus (2)(I) Precalculus (2)(O) 5.08 Inverse Trigonometric Functions and Graphs Precalculus (2)(F) Precalculus (2)(H) Precalculus (2)(I) Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

Section 5: Introduction to Trigonometry and Graphs · Section 5: Introduction to Trigonometry and Graphs ... , tangent, secant, cosecant, and cotangent. ... Graphs of Tangent and

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Section5:IntroductiontoTrigonometryandGraphsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).5.01RadiansandDegreeMeasurements

• Precalculus(4)(B)• Precalculus(4)(C)• Precalculus(4)(D)

5.02LinearandAngularVelocity

• Precalculus(4)(D)5.03TrigonometricRatios

• Precalculus(4)(E)5.04TrigonometricAnglesandtheUnitCircle

• Precalculus(2)(P)• Precalculus(4)(A)

5.05GraphsofSineandCosine

• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(O)

5.06GraphsofSecantandCosecant

• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(O)

5.07GraphsofTangentandCotangent

• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(O)

5.08InverseTrigonometricFunctionsandGraphs

• Precalculus(2)(F)• Precalculus(2)(H)• Precalculus(2)(I)

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5.01RadianandDegreeMeasurements

Trigonometryisthemeasureoftriangles.

Angle–Tworayswithacommonendpoint

• AnangleisusuallydenotedwiththeGreekletter𝜃,pronounced“theta.”

• Oneoftheraysistheinitialsideandtypicallystartsonthepositive𝑥-axisat0°(alsoknownasstandardposition).

• Theotherrayisknownastheterminalside.

• Youwillencountermanytypesofangles,includingthefollowing:

o Positiveangles–Rotatecounterclockwise

o Negativeangles–Rotateclockwiseo Quadrantalangles–The𝑥-and𝑦-axis

angleso Coterminalangles–Twoangleswiththe

sameinitialandterminalsideso Complementaryangles–Twoacute

angleswhosesumis90°o Supplementaryangles–Twopositive

angleswhosesumis180°

Themeasurementofanangleisbasedontheamountofrotationfromtheinitialsidetotheterminalside.Twocommonunitsofmeasureusedforanglesaredegreesandradians.

Howmanydegreesareinacircle?

Whatistheequationforthecircumferenceofacircle?

Usingthesetwoanswerswecanfindaconversionbetweendegreesandradians:

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1. Convert150°toradians.

2. Findthequadrantinwhichananglemeasuring420°islocated.

3. Findthequadrantinwhichananglemeasuring3radiansislocated.

4. Findthesmallestpositivecoterminalanglefortheangle 176p

- .

5. Whichpairofanglesiscomplementary?i. 10°and80°ii. −10°and100°

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Weuseradianswhenmeasuringarclengthandtheareaofasectorofacircle.

Arclength–Definedas𝑠 = 𝑟𝜃

Areaofasector–Definedas𝐴 = )*𝑟*𝜃

• Bothequationsarederivedfromthecircumferenceandareaofacircle.• Inbothequations,thecentralangle,𝜃,ismeasuredinradians.

6. Acirclewitharadiusof3incheshasanarclengthof6inches.Findthecentralangleinradians.

7. Acirclehasadiameterof12feetandacentralangleof40°.Whatistheareaofthesector?

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5.02AngularandLinearVelocity

Consideranobjectmovinginacircularpathataconstantspeed.Thedistance,𝑠,ittravelsalongthecircleinagivenperiodoftime,𝑡,isthelinearspeed,𝑣.Theangle(inradians)inwhichthesameobjectrotatesinagivenperiodoftimeistheangularspeed,𝜔(pronounced“omega”).

Theequationsassociatedwithlinearspeedandangularspeedare𝑣 = ./and𝜔 = 0

/,

respectively.

Asidefromtheequationsforangularandlinearspeed,anotherwaytoconvertbetweenthesespeedsistouseunitconversions.

1. Findtheangularspeed,inrevolutionsperminute(rpm),ofatopwitharadiusof2centimetersthatisspinning20cm/min.

2. Amerry-go-roundwithadiameterof6feetspinsatanangleof7radiansin2seconds.Findtheangularspeedinrevolutionsperminute.

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5.03TrigonometricRatios

Oneapplicationoftrigonometryistorelatetheratiooftwosidesofarighttriangle.Thisproducessixcombinations:sine,cosine,tangent,secant,cosecant,andcotangent.

Givenanacuteangle𝐴fromstandardposition,wedefinethesixtrigonometricratiosasfollows:

sin 𝐴 = 45 cos 𝐴 = 8

5 tan𝐴 = 4

8

csc 𝐴 = 54 sec 𝐴 = 5

8 cot 𝐴 = 8

4

Amoregeneralruleforthetrigonometricratiosrelatesthemtotheoppositeside,adjacentside,andhypotenuseofarighttriangle:

sin 𝐴 = <==<>?@ABC=<@ADE>A

cos 𝐴 = FGHFIAD@BC=<@ADE>A

tan𝐴 = <==<>?@AFGHFIAD@

csc 𝐴 = BC=<@ADE>A<==<>?@A

sec 𝐴 = BC=<@ADE>AFGHFIAD@

cot 𝐴 = FGHFIAD@<==<>?@A

Andsincewearediscussingrighttriangles,recallthePythagoreantheorem𝑥* + 𝑦* = 𝑟*.

Thequadranttheangleisinwilldeterminewhetherthetrigonometricratioispositiveornegative.

OnewaytorememberthesignofvarioustrigonometricratiosisbyusingthephraseAllStudentsTakeCalculus.

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Whengivenonetrigonometricratioandaskedtofindanother,thestepstosolveareasfollows:

1) Calculate𝑥,𝑦,and𝑟.2) Calculatetheothertrigonometricratio.3) Doublecheckwhetheryourfinalanswerispositiveornegativebased

onthequadrantitisin.

1. Findcos𝐴ifsin𝐴 = KLand𝐴isinquadrantII.

2. Findsin𝐵iftan𝐵 = − OLandsec 𝐵 > 0.

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5.04TrigonometricAnglesandtheUnitCircle

Let’sstartbyexploringsomecommontrianglesfromgeometry.

Isoscelesrighttriangleshavetwocongruentsideswithcorrespondinganglesof45°.Wecanfindthesineandcosineof45°asfollows:

Equilateraltriangleshavethreecongruentsideswiththreecongruentanglesof60°.Bisectinganequilateraltrianglewillgiveustwotriangleswithanglesof30°,60°,and90°.Wecanthenfindsineandcosineof30°and60°asfollows:

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Oneapproachtorememberingtheseanglesistheunitcircle.

Anotherapproachisbyorganizingtheanglesinachart:

Degrees 𝟎𝒐 𝟑𝟎𝒐 𝟒𝟓𝒐 𝟔𝟎𝒐 𝟗𝟎𝒐

Radians 𝟎𝝅𝟔

𝝅𝟒

𝝅𝟑

𝝅𝟐

sin 𝜃

cos 𝜃

Now,tofindtheotherfourtrigonometricratios,wemustconsidertheidentitiesoftheremainingtrigonometricfunctionsinrelationtothesineandcosine:

csc 𝜃 = )>?D0

sec 𝜃 = )I<>0

tan 𝜃 = >?D0I<>0

cot 𝜃 = I<>0>?D 0

Remember:UseAllStudentsTakeCalculustodeterminewhetherthetrigonometricratioispositiveornegative,andusethereferenceangletodeterminewhichquadrantyouarein.

Referenceangle–Theacuteangleformedbetweentheterminalsideandthehorizontalaxis

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1. Evaluatecos225°.

2. Evaluatecsc(– 120°).

3. Evaluatesin ObK.

4. Evaluatesec− ObL.

5. Evaluatecot )cbd.

6. Supposean8-footladderleaningagainstawallmakesanangleof30°withthewall.Howfaristhebaseoftheladderfromthewall?

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5.05GraphsofSineandCosine

𝑦 = sin𝑥 𝑦 = cos𝑥

Keypoints–Fivepointsononeperiodofatrigonometricfunctiongraph

Overall,asineandcosinefunctionwillhavetheforms𝑦 = 𝑑 + 𝑎sin(𝑏𝑥 − 𝑐)and

𝑦 = 𝑑 + 𝑎cos(𝑏𝑥 − 𝑐).

• Amplitude–Theheightofthewave,representedby𝑎

o Theamplitude|a|ishalfthedistancefromthemaximumtominimumy-value.

o Fromthecenter,youwouldgoupordown|a|unitstofindthecrestortrough,respectively.

• Period–Onecycleofthewave,equalto*kl

o sin 𝑥 + 2𝜋 = sin 𝑥o cos 𝑥 + 2𝜋 = cos 𝑥

• Verticaltranslation–Shiftup(ifpositive)ordown(ifnegative),representedby𝑑

• Phaseshift–Shiftleft(+)orright(−),equaltono

o Thephaseshiftisthestartinglocationforoneperiodofatrigonometricfunction.

o Tofindthephaseshift,settheinsideequaltozeroandsolvefor𝑥.

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GraphingTrigonometricFunctionsUsingtheBoxMethod

Tographatrigonometricfunction,wearegoingtoemploythe“boxmethod”:

A. Figureouttheamplitude,period,andshifts.

Step1: Startat(0,0).Moveupordownthenecessaryverticaltranslationanddrawahorizontaldottedline.Thisisthemiddleofthebox.

Step2: Fromthehorizontallineyoujustdrew,moveupanddownthegivenamplitudeanddrawhorizontaldottedlinesatthesepoints.Thesearethetopandbottomofthebox.

Step3: Fromthepoint(0,0),goleftorrightthenecessaryphaseshiftanddrawaverticalline.Thisisthestartofthebox.

Step4: Fromtheverticallineyoujustdrew,gooveroneperiodanddrawanotherverticalline.Thisistheendofthebox.(Inotherwords,youareaddingthelocationofthephaseshiftplustheperiod.)

B. Drawthedesiredwaveinsidetheboxyouhavedrawn.

𝑦 = sin𝑥 𝑦 = −sin𝑥

𝑦 = cos𝑥 𝑦 = −cos𝑥

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1. Findthefivekeypointsof𝑦 = 3 − 2 sin 𝑥andgraphoneperiodoftheequation.

2. Thepistonsinacaroscillateupanddownlikeasinewave.Ifapistonmoves0.42inchesfromthebottomtothetopofitsoscillationin0.3seconds,determinetheequationforthemotionofthepiston’sheight(ininches)intermsoftime(inseconds).Assumethatat𝑡 = 0thepistonstartsinthemiddle(𝑦 = 0).

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5.06GraphsofSecantandCosecant

𝑦 = sec𝑥 = )n<>8

𝑦 = csc𝑥 = )>?D8

Thegeneralequations𝑦 = 𝑑 + 𝑎sec(𝑏𝑥– 𝑐)and𝑦 = 𝑑 + 𝑎csc(𝑏𝑥– 𝑐)havetwonewcharacteristicsinadditiontothecharacteristicsalreadydescribedforsineandcosinefunctions.Thesenewcharacteristicsareverticalasymptotesanddomainrestrictions.

Verticalasymptotes–Verticaldashedlinesthatmarkthelimitsofafunctiongraph,sothatacurvemayapproachtheasymptotesbutneverintersectthem

• Verticalasymptotesof𝑦 = 𝑐sc 𝑥arelocatedat𝑥 = 𝜋𝑛, 𝑛 = 0,±1,±2,…

• Verticalasymptotesof𝑦 = sec𝑥arelocatedat𝑥 = b*+ 𝜋𝑛, 𝑛 = 0,±1,±2,…

Domainrestrictions–Inputsorpointsthatareexcludedwhenevaluatingafunction

1. Findthefivekeypointsof𝑦 = 2 +)*cos(2𝑥– 𝜋)andgraphoneperiodoftheequation.

Usethisgraphtothengraphoneperiodof𝑦 = 2 +)*sec(2𝑥– 𝜋).

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5.07GraphsofTangentandCotangent

𝑦 = tan𝑥 𝑦 = cot𝑥

Thegeneralequations𝑦 = 𝑑 + 𝑎 tan(𝑏𝑥 − 𝑐)and𝑦 = 𝑑 + 𝑎 cot(𝑏𝑥 − 𝑐)havethefollowingcharacteristics:

• Theperiodfortangentandcotangentisbo.

o tan 𝑥 + 𝜋 = tan 𝑥o cot 𝑥 + 𝜋 = cot 𝑥

• Tangentandcotangentfunctionshaveverticalasymptotesanddomainrestrictionsasfollows:

o Verticalasymptotesof𝑦 = cot 𝑥arelocatedat𝑥 = 𝜋𝑛, 𝑛 = 0,±1,±2,…

o Verticalasymptotesof𝑦 = tan𝑥arelocatedat𝑥 = b*+ 𝜋𝑛, 𝑛 = 0,±1,±2,…

Tofindtwoasymptotesofatrigonometricfunctiongraph,settheinsideequaltotwoconsecutiveasymptotesoftheoriginalgraph.

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1. Graphoneperiodof𝑦 = 3 tan 3𝑥 − 𝜋 .

2. Graphtwoperiodsof𝑦 = 3 cot 𝜋𝑥 − bL.

3. Findalltheverticalasymptotesof𝑦 = −2 tan 𝑥 − bL.

4. Findallthe𝑥-interceptsof𝑦 = 2 cot 4𝑥 + 𝜋 .

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5.08InverseTrigonometry

Recallthataninversefunctionmustpassthehorizontallinetest:

Ifwegraph𝑦 = sin𝑥,wehavethefollowing:

Sometimesweneedtorestrictthedomainof𝑓(𝑥)toobtainaninverse.

For𝑦 = sin𝑥,wecanrestrictitasfollows:

Andwecangraphtheinverse𝑦 = arcsin𝑥,alsowrittenas𝑦 = sinx)𝑥:

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For𝑦 = cos𝑥,wecanrestrictitasfollows:

Andwecangraphtheinverse𝑦 = arccos𝑥,alsowrittenas𝑦 = cosx)𝑥:

SummaryofInverseFunctions

Thefunctions𝑦 = arcsin 𝑥,𝑦 = arccsc 𝑥,and𝑦 = arctan 𝑥existinquadrantsIandIVinreferencetotheunitcircle.

Thefunctions𝑦 = arccos 𝑥,𝑦 = 𝑎𝑟𝑐𝑠𝑒𝑐 𝑥,and𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑡 𝑥existinquadrantsIandIIinreferencetotheunitcircle.

Thesepropertiesapplyto𝑦 = arcsin 𝑥,𝑦 = arccsc 𝑥,and𝑦 = arctan 𝑥:

• Ifyoutaketheinverseofapositivenumber,itisinquadrantI.

• Ifyoutaketheinverseofanegativenumber,itisinquadrantIV.

Thesepropertiesapplyto𝑦 = arccos 𝑥,𝑦 = arcsec 𝑥,and𝑦 = arccot 𝑥:

• Ifyoutaketheinverseofapositivenumber,itisinquadrantI.

• Ifyoutaketheinverseofanegativenumber,itisinquadrantII.

Rememberdomainandrangerestrictionsineverycase.

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1. Evaluateeachinversetrigonometricfunction:

i. sinx) )*

ii. sinx) − )*

iii. tanx) 1

iv. tanx) −1

v. cosx) )*

vi. cosx) − )*

2. Evaluateeachinversetrigonometricfunction:

i. sinx) sin {bd

ii. cosx) cos {bd

iii. tanx) sin 𝜋

3. Evaluateeachtrigonometricfunction:

i. sin arccos − )K

ii. cos arctan −2

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