7.4 Off on a Tangent - uen.org · segment AC as O=cos (;), segment BC as B=sin(;) and segment AB as r = 1. In the second drawing note that DABC is similar to DADE and that the measure

SECONDARY MATH III // MODULE 7

TRIGONOMETRIC FUNCTIONS, EQUATIONS & IDENTITIES – 7.4

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

7.4 Off on a Tangent

A Develop and Solidify

Understanding Task

Recallthattherighttriangledefinitionofthetangentratiois:

tan(%) =()*+,-/0123)/44/12,)5*+()6

()*+,-/0123)53758)*,,/5*+()6

1. Revisethisdefinitiontofindthetangentofanyangleofrotation,givenineitherradiansordegrees.Explainwhyyourdefinitionisreasonable.

2. Revisethisdefinitiontofindthetangentofanyangleofrotationdrawninstandardpositionontheunitcircle.Explainwhyyourdefinitionisreasonable.

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Wehaveobservedthatontheunitcirclethevalueofsineandcosinecanberepresented

withthelengthofalinesegment.

3. Indicateonthefollowingdiagramwhichsegment’slengthrepresentsthevalueofsin(;)andwhichrepresentsthevalueofcos(;)forthegivenangleq.

Thereisalsoalinesegmentthatcanbedefinedontheunitcirclesothatitslengthrepresentsthevalueoftan(;).Considerthelengthof?@AAAAintheunitcirclediagrambelow.

NotethatDADEandDABCarerighttriangles.WriteaconvincingargumentexplainingwhythelengthofsegmentDEisequivalenttothevalueoftan(;)forthegivenangleq.

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4. OnthecoordinateaxesbelowsketchthegraphofB = tan(;)byconsideringthelengthof

segmentDEasqrotatesthroughanglesfrom0radiansto2pradians.Explainanyinterestingfeaturesyounoticeinyourgraph.

ExtendyourgraphofB = tan(;)byconsideringthelengthofsegmentDEasqrotatesthroughnegativeanglesfrom0radiansto-2pradians.

5. UsingyourunitcirclediagramsfromthetaskWaterWheelsandtheUnitCircle,giveexactvaluesforthefollowingtrigonometricexpressions:

a.tan CD

EF = b.tan C

GD

EF = c.tan C

HD

EF =

d.tan CD

IF = e.tan C

JD

IF = f.tan C

KKD

EF =

g.tan CD

LF = h.tan(M) = i.tan C

HD

JF =

23




Functionsareoftenclassifiedbasedonthefollowingdefinitions:

•AfunctionN(O)isclassifiedasanoddfunctionifN(−;) = −N(;)

•AfunctionN(O)isclassifiedasanevenfunctionifN(−;) = N(;)

6. Basedonthesedefinitionsandyourworkinthismodule,determinehowtoclassifyeachofthefollowingtrigonometricfunctions.

• ThefunctionB = QRS(O)wouldbeclassifiedasan[oddfunction,evenfunction,neitheranoddorevenfunction].Giveevidenceforyourresponse.

• ThefunctionB = TUQ(O)wouldbeclassifiedasan[oddfunction,evenfunction,neitheranoddorevenfunction].Giveevidenceforyourresponse.

• ThefunctionB = VWS(O)wouldbeclassifiedasan[oddfunction,evenfunction,neitheranoddorevenfunction].Giveevidenceforyourresponse.

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7.4 Off on a Tangent – Teacher Notes A Develop and Solidify Understanding Task

Purpose:Thepurposeofthistaskistoextendthedefinitionofthetangentfromtherighttriangle

trigonometricratiodefinition,tan(%) =XYSZV-/0123)/44/12,)5*+()6()*+,-/0123)53758)*,,/5*+()6

,toanangleofrotation

definition:tan(;) =[

\.Thegraphofthetangentfunctionisobtainedbyrepresentingthetangent

ofanangleofrotationbythelengthofalinesegmentrelatedtotheunitcircle,andtrackingthe

lengthofthelinesegmentastheangleofrotationincreasesaroundtheunitcircle.The

trigonometricidentitytan(;) =]^_(`)

ab](`)isalsoexploredintermsoftheunitcircle.

CoreStandardsFocus:

F.TF.2Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometric

functionstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwise

aroundtheunitcircle.

F.TF.3(+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentfor

π/3,π/4andπ/6,andusetheunitcircletoexpressthevaluesofsine,cosine,andtangentforπ–x,

π+x,and2π–xintermsoftheirvaluesforx,wherexisanyrealnumber.

F.TF.4(+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometric

functions.

F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative

relationshipitdescribes.

RelatedStandards:F.IF.4,F.IF.7,F.IF.9




StandardsforMathematicalPractice:

SMP7–Useappropriatetoolsstrategically

Vocabulary:Studentswilldefinethetangentfunctionforanangleofrotationastan ; = [

\wherex

andyarethecoordinatesofapointonacirclewheretheterminalrayoftheangleofrotation

intersectsthecirclewhentheangleisdrawninstandardposition(i.e.,thevertexoftheangleisat

theoriginandtheinitialrayliesalongthepositivex-axis.)

TheTeachingCycle:

Launch(WholeClass):

Remindstudentsthatwehaveredefinedsineandcosineforanglesofrotationdrawninstandard

positionbyusingthevaluesofx,yandr.Askhowtheymightredefinetangentusingthesesame

values.Studentsshouldnotethatthedefinitiontan(;) =[

\isindependentofthevalueofr.

Examinethetwounitcircledrawingsinquestion3togetherasaclass.Inthedrawingslabel

segmentACasO = cos(;),segmentBCasB = sin(;)andsegmentABasr=1.Inthesecond

drawingnotethatDABCissimilartoDADEandthatthemeasureofsegmentAEis1.Usingthis

informationaskstudentstoconsiderwhatthisimpliesaboutthemeasureofsegmentDE.Give

studentsacoupleofminutestosuggestthatsincethetrianglesaresimilartheycanwritethe

proportioncd

6d=

ef

6for

cd

K=

[

\.TheyshouldrecognizethatthelengthofsegmentDEisdefinedin

thesamewaythatwehavedefinedtan(;).Thatis,thelengthofsegmentDErepresentsthevalueof

tan(;)inthesamewaythatthelengthofsegmentACrepresentsthevalueofcos(;)andthelength

ofsegmentBCrepresentsthevalueofsin(;).Youmayalsowanttopointoutthatthetrigonometric

identitytan(;) =]^_(`)

ab](`)ispresentinthisdiagram.

Nowthatwehaveawayofvisuallyrepresentingthemagnitudeofthevalueoftan(;),assign

studentstoworkondeterminingwhatthisimpliesabouttheshapeandfeaturesofthegraphof

y = tan(;).Alsohavethemworkontherestofthetaskbyusingtheirunitcirclediagrams.




Explore(SmallGroup):

Ifstudentsarehavingahardtimesketchingthegraph,focustheirattentiononsmallintervalsofq.

Forexample,whathappenstothelengthofsegmentDEasqincreasesfrom0radianstoDLradians?

Whathappenswhen; =D

L?WhathappenswhenqincreasesfromD

Ltop?Howwouldyoudraw

DADEonthisinterval?Whataboutnegativeanglesofrotation?

Watchasstudentscomputevaluesoftan(;)usinginformationrecordedontheirunitcircle

diagrams.Studentsmayneedhelpsimplifyingtheratiosformedby[

\.Allowstudentstoleave

theseratiosunsimplifieduntilthewholeclassdiscussionwhenyoucandiscusssomeofthe

arithmeticinvolved,hopefullybyusingworkfromstudentswhoaresuccessfulatsimplifyingthese

ratios.Lookforsuchstudents.

Listenforhowstudentsapplythedefinitionsofoddandevenfunctionstothesine,cosineand

tangentfunctions.Whatrepresentationsdotheydrawupontomakethesedecisions:thesymmetry

ofpointsaroundtheunitcircle,agraphofthefunction,orsomeotherwaysofreasoning?

Discuss(WholeClass):

Focusthewholeclassdiscussiononthefollowingthreeitems:

• Thegraphofthetangentfunction,includingtheperiodofpandthebehaviorofthegraph

nearandat±D

Land±

JD

L(theverticalasymptotes).

• ThevaluesofthetangentfunctionatanglesthataremultiplesofD

Eand

D

I,includingthe

arithmeticofsimplifyingtheseratios.

• Theclassificationofsine,cosineandtangentasevenoroddfunctionsandtheevidenceused

tosupporttheseclassifications(e.g.,thegraphofthefunctionorthesymmetryoftheunit

circle).

AlignedReady,Set,Go:TrigonometricFunctions,EquationsandIdentities7.4


TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4


7.4

Needhelp?Visitwww.rsgsupport.org

READY Topic:Makingrigidandnon-rigidtransformationsonfunctions

Theequationofaparentfunctionisgiven.Writeanewequationwiththegiventransformations.Thensketchthenewfunctiononthesamegraphastheparentfunction.(Ifthefunctionhasasymptotes,sketchthemin.)1.$ = &'

Verticalshift:up8horizontalshift:left3dilation:¼Equation:Domain:Range:

2.$ = )

*

Verticalshift:up4horizontalshift:right3dilation:−1Equation:Domain:Range:

READY, SET, GO! Name PeriodDate

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7.4


3.$ = √&

Verticalshift:none.horizontalshift:left5dilation:3

Equation:Domain:Range:

4.$ = sin &

Verticalshift:1

horizontalshift:left0'

dilation(amplitude):3

Equation:

Domain:

Range:

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7.4


SET Topic:Connectingvaluesinthespecialtriangleswithradianmeasures

5.TriangleABCisarighttriangle.AB=1.

Usetheinformationinthefiguretolabelthelengthofthesidesandmeasureoftheangles.

6.TriangleRSTisanequilateraltriangle.RS=1123333isanaltitude

Usetheinformationinthefiguretolabelthelengthofthesides,thelengthof423333,andtheexactlengthof123333.

LabelthemeasureofanglesRSAandSRA.

7.Usewhatyouknowabouttheunitcircleandtheinformationfromthefiguresinproblems

5and6tofillinthetable.Somevalueswillbeundefined.

function 5 =6

6 5 =

6

4 5 =

6

3 5 =

6

2 5 = 6

5 =36

2 5 = 26

sin 5

cos 5

tan 5

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7.4


8.Labelallofthepointsandanglesofrotationinthegivenunitcircle.

9.Graph?(A) = CDE F.UseyourtableofvaluesaboveforG(&) = tan5.Sketchyourasymptoteswithdottedlines.

10.Wheredoasymptotesalwaysoccur?

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7.4


GO Topic:Recallingtrigfacts

Answerthequestionsbelow.Besureyoucanjustifyyourthinking.

11.GiventriangleABCwithangleCbeingtherightangle,whatisthesumofH∠2+H∠K?

12.Identifythequadrantsinwhichsin 5MNOPNMQMRS.

13.Identifythequadrantsinwhichcos 5MNTSUVQMRS.

14.Identifythequadrantsinwhichtan 5MNOPNMQMRS.

15.Explainwhyitisimpossibleforsin5 > 1.

16.Nametheanglesofrotation(inradians)forwhensin 5 = cos 5.

17.Forwhichtrigfunctionsdoapositiverotationandanegativerotationalwaysgivethesamevalue?

18.Explainwhyintheunitcircletan 5 = X*.

19.Whichfunctionconnectswiththeslopeofthehypotenuseinarighttriangle?

20.Explainwhysin5=cos(90°−5).�

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Documents

7.4 Off on a Tangent - uen.org · segment AC as O=cos (;), segment BC as B=sin(;) and segment AB as r = 1. In the second drawing note that DABC is similar to DADE and that the measure