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

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


5.07GraphsofTangentandCotangent


5.08InverseTrigonometricFunctionsandGraphs

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

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

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:


1. Convert150°toradians.

2. Findthequadrantinwhichananglemeasuring420°islocated.

3. Findthequadrantinwhichananglemeasuring3radiansislocated.

4. Findthesmallestpositivecoterminalanglefortheangle 176p

- .

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


Weuseradianswhenmeasuringarclengthandtheareaofasectorofacircle.

Arclength–Definedas𝑠 = 𝑟𝜃

Areaofasector–Definedas𝐴 = )*𝑟*𝜃

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

6. Acirclewitharadiusof3incheshasanarclengthof6inches.Findthecentralangleinradians.

7. Acirclehasadiameterof12feetandacentralangleof40°.Whatistheareaofthesector?


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.


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.


Whengivenonetrigonometricratioandaskedtofindanother,thestepstosolveareasfollows:

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

onthequadrantitisin.

1. Findcos𝐴ifsin𝐴 = KLand𝐴isinquadrantII.

2. Findsin𝐵iftan𝐵 = − OLandsec 𝐵 > 0.


5.04TrigonometricAnglesandtheUnitCircle

Let’sstartbyexploringsomecommontrianglesfromgeometry.

Isoscelesrighttriangleshavetwocongruentsideswithcorrespondinganglesof45°.Wecanfindthesineandcosineof45°asfollows:

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


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


1. Evaluatecos225°.

2. Evaluatecsc(– 120°).

3. Evaluatesin ObK.

4. Evaluatesec− ObL.

5. Evaluatecot )cbd.

6. Supposean8-footladderleaningagainstawallmakesanangleof30°withthewall.Howfaristhebaseoftheladderfromthewall?


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.


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


1. Findthefivekeypointsof𝑦 = 3 − 2 sin 𝑥andgraphoneperiodoftheequation.

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


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).


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.


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 .


5.08InverseTrigonometry

Recallthataninversefunctionmustpassthehorizontallinetest:

Ifwegraph𝑦 = sin𝑥,wehavethefollowing:

Sometimesweneedtorestrictthedomainof𝑓(𝑥)toobtainaninverse.

For𝑦 = sin𝑥,wecanrestrictitasfollows:

Andwecangraphtheinverse𝑦 = arcsin𝑥,alsowrittenas𝑦 = sinx)𝑥:


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.


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