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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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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
Wehaveobservedthatontheunitcirclethevalueofsineandcosinecanberepresented
withthelengthofalinesegment.
3. Indicateonthefollowingdiagramwhichsegment’slengthrepresentsthevalueofsin(;)andwhichrepresentsthevalueofcos(;)forthegivenangleq.
Thereisalsoalinesegmentthatcanbedefinedontheunitcirclesothatitslengthrepresentsthevalueoftan(;).Considerthelengthof?@AAAAintheunitcirclediagrambelow.
NotethatDADEandDABCarerighttriangles.WriteaconvincingargumentexplainingwhythelengthofsegmentDEisequivalenttothevalueoftan(;)forthegivenangleq.
22
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
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
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
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.
24
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 – 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
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
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.
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
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
SECONDARY MATH III // MODULE 7
TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
25
SECONDARY MATH III // MODULE 7
TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.4
Needhelp?Visitwww.rsgsupport.org
3.$ = √&
Verticalshift:none.horizontalshift:left5dilation:3
Equation:Domain:Range:
4.$ = sin &
Verticalshift:1
horizontalshift:left0'
dilation(amplitude):3
Equation:
Domain:
Range:
26
SECONDARY MATH III // MODULE 7
TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.4
Needhelp?Visitwww.rsgsupport.org
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
27
SECONDARY MATH III // MODULE 7
TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.4
Needhelp?Visitwww.rsgsupport.org
8.Labelallofthepointsandanglesofrotationinthegivenunitcircle.
9.Graph?(A) = CDE F.UseyourtableofvaluesaboveforG(&) = tan5.Sketchyourasymptoteswithdottedlines.
10.Wheredoasymptotesalwaysoccur?
28
SECONDARY MATH III // MODULE 7
TRIGONOMETRIC FUNCTIONS, EQUATIONS, AND IDENTITIES – 7.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.4
Needhelp?Visitwww.rsgsupport.org
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).�
29
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