angles and ratios - engage explore inspire - …...mcr3u – unit 5: trigonometric ratios – lesson...
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MCR3U–Unit5:TrigonometricRatios–Lesson1 Date:___________
Learninggoal:Icanfindco-terminalandrelatedacuteangles.Icancalculatetrigandreciprocaltrigratios.
AnglesandRatios
ANGLESINSTANDARDPOSITION
Anangleinstandardpositionhasitsvertexattheoriginanditsinitialarmonthepositivex-axisofacoordinate
gridandrotatescounterclockwise(up).
Thecoordinategridisbrokeninto4quadrants.
Example1:Sketcheachangleinstandardpositionandlabelthequadrantitisin.
a) 140° b)240° c)–60° d)–210°
CO-TERMINALANGLES
Co-terminalanglesareanglesinstandardpositionthathavethesameterminalarm.Therearemanywaysto
arriveatthesameterminalarm.Tofindco-terminalanglesyouaddorsubtractmultiplesof360°.Example2:Sketcheachangleinstandardpositionandlabelitsquadrant.Then,sketchananglethatisco-terminal.
a)60° b)200° c)–30° d)–190°
RELATEDACUTEANGLES
Therelatedacuteangleistheanglebetweentheterminalarmofanangleinstandardpositionandthex-axis
whentheterminalarmisinquadrantII,III,orIV.Thisangleisalwaysbetween0oand90
oandisalways
positive.
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y
x
Example3:Sketcheachangleinstandardposition.Then,sketchandidentifytherelatedacuteangle.
a)160° b)30° c)280° d)205° d)-10°
PRIMARYTRIGRATIOS
PrimarytrigratioscanONLYbeusedinrightangletriangles!Makesureyourcalculatorisindegreemode.Adegreeisone360thofacircle.Thethreeprimarytrigratiosarelikeoperationsandfunctions,buttheyarecalledratiosbecausetheymeasuretheratiosofsidesrelatedtoangles.Wetypicallysay“sineofangleA”or“sineofA”.
!"#$ = !""!#$%&
!!"#$%&'(%
!"#$ = !"#!$%&'!!"#$%&'(%
!"#$ = !""!#$%&!"#!$%&'
Inthetriangleabove….
!"#$ = !"#$ = !"#$ =
Example4:Solveforthemissingside,!.a) b)
A
B
C θ
(hyp) c
a (opp)
b (adj)
SOH CAH TOA
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30cm
SOLVINGFORANANGLETosolveforananglewhentheratioisknown,wemusttaketheinversetrigoperationofbothsides.
Example4:Solvefortheindicatedangletothenearestdegree.
a) b)
RECIPROCALTRIGFUNCTIONS
Acoupleunitsbackwelookedattheinverseandreciprocalforfunction,! ! .Wenoticedtherewasadifferentbetweenthetwo.Thereciprocalfunctionfor!(!)isfunction!(!) = !
!(!).Justlikethepreviousfunctionswe
haveseen,thereisadifferencebetweentheinverseofatrigfunctionandareciprocaltrigfunction.
Wecanundotheseoperationswithinversesjustlikewedidwiththethreeprimarytrigratios.
NOTE:Calculatorsdon’thavebuttonsfortheseratios.Itisbesttotranslatefromreciprocalratiostoprimaryratiosandthensolve.
Operation !"# !"# !"#
InverseOperation
Operation !"! !"# !"#
InverseOperation
Function !"#$ !"#$ !"#$
ReciprocalFunctionCosecant(csc)
!"!# = 1!"#$
Secant(sec)
!"#$ = 1!"#$
Cotangent(cot)
!"#$ = 1!"#$
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Example5:Evaluatetothenearestthousandth.
a)!"#54° b)!"#25° c)!"!85°
Example6:Determinethevalueof!tothenearestdegree.a)!"#$ = 2.4752 b)!"!# = 1.4945 c)!"#$ = 3.8637
HW:TrigRatiosWorksheet,Pg.422#1-3,7-9anscorr(8h1690)
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TrigRatiosWorksheet
1. Given∆!"#,statethe3primaryand3reciprocaltrigonometricratiosfor∠!.
2. Statethereciprocaltrigonometricratiosthatcorrespondto:
a)!"#$ = !!" b)!"#$ = !"
!" c) !"#$ = !!"
3. Foreachprimarytrigonometricratio,determinethecorrespondingreciprocalratio.
a)!"#$ = !! b)!"#$ = !
! c) !"#$ = !! c) !"#$ = !
!
4. Evaluatetothenearestthousandth. a)!"#34° b)!"#10° c) !"#75° c) !"!45°
5. a)Foreachtriangle,calculate!"!#,!"#$,and!"#$. b)Foreachtriangle,useoneofthereciprocalratiostodetermine!.
i) ii) iii) iv)
6. Determinethevalueof!tothenearestdegree.a)!"#$ = 3.2404 b)!"!# = 1.2711 c)!"#$ = 1.4526 d)!"#$ = 0.5814
7. Givenanyrighttrianglewithanacuteangle!, a)explainwhy !"#$isalwayslessthanorequalto1. b)explainwhy!"!#isalwaysgreaterthanorequalto1.
12 cm
5 cm13 cm
AC
B
8
610
8.5
8.5
12
2
33.6
8
15
17
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TrigRatiosWorksheetAnswers
1. 135sin =A 13
12cos =A 12
5tan =A
513csc =A 12
13sec =A 512
cot =A
2. 817csc =σ 15
17sec =σ 815
cot =σ
3.a) 34sec =σ b) 3
2cot =σ c) 2csc =σ d) 4cot =σ
4.a)0.829 b)1.015 c)0.268 d)1.414
5.i) a) 35
610csc ==σ 4
5810sec ==σ 3
468
cot ==σ b) o9.36=σ
ii) a) 1724
5.812csc ==σ 17
245.812sec ==σ 1
5.85.8
cot ==σ b) o45=σ
iii) a) 56
3036
36.3csc ===σ 5
92036
26.3sec ===σ 3
2cot =σ b) o56=σ
iv) a) 817csc =σ 15
17sec =σ 815
cot =σ b) o28=σ
6.a) o17=σ b) o52=σ c) o46=σ d) o60=σ
7.a) hypadj
=σcos ,andthehypotenuseislongerthantheadjacent,bydefinition.
b) opphyp
=σcsc ,andthehypotenuseislongerthantheopposite,bydefinition.
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MCR3U–Unit5:TrigonometricRatios–Lesson2 Date:___________Learninggoal:Icandefineandcalculatetrigratiosandanglesbasedonacoordinatepointontheterminalarm.
CASTRuleHowareweabletoevaluatethesineof225°if225°cannotbethecornerofatriangle?TheStandardPositionofanglesallowsustodefinetrigonometricratiosforANYangle.
Tofindthetrigratiosfor!,pickapoint!(!,!)on ! theterminalarm.Dropaverticallinetothe !-axis!toconstructarighttriangle. !"#$ = !"#$ = !"# ! =
INVESTIGATION Calculateeachprimarytrigratio.Leaveyouranswerinexactform.
!"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = !"#$ = SUMMARYLookverycarefullyatthesignsofyourtrigratiosineachquadrant.Inthefollowingquadrantsystemindicatewhetherthetrigratiowaspositive(+)ornegative(-)forthefourdifferentquadrantsinvestigatedabove.
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Example1:Solvefor!if0 ≤ ! ≤ 360°.a)!"#$ = !
! b) !"#$ = −1
Example2:Thepoint(−5,12)liesontheterminalarmofangle !instandardposition.a)Determinetheprimarytrigonometricratiosforangle!.
b)Determinethereciprocaltrigratiosforangle !.
c)Calculatethevalueof!tothenearestdegree.
Example3:Giventhefollowingprimarytrigratiosfindallpossibleexactvaluesoftheothertwotrigratiosfor
a)!"#$ = − !! ,! liesinquadrantIII b) !"#$ = !
!,0 ≤ ! ≤ 360°.
HW:CASTRuleWorksheet
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CASTRuleWorksheet
1. Foreachtrigonometricratio,useasketchtodetermineinwhichquadranttheterminalarmoftheprincipalanglelies,thevalueoftherelatedacuteangle β ,andthesignoftheratio.
a) o315sin b) o110tan c) o285cos d) o225tan
2. Eachpointliesontheterminalarmofangleσ instandardposition.
i)Determinetheprimarytrigonometricratiosforangleσ .
ii)Calculatethevalueofσ tothenearestdegree.
a) ( )11,5 b) ( )3,8− c) ( )8,5 −− d) ( )8,6 −
3. UsingaCartesianplane,explainwhy…
a) 0180sin =o b) 1180cos −=o c) 0180tan =o
d)2145sin =o e) oo 45sin45cos = f) 145tan =o
4. Useeachtrigonometricratiotodetermineallvaluesofσ ,tothenearestdegreeif oo 3600 ≤≤ σ .
a) 4815.0sin =σ b) 1623.0tan −=σ c) 8722.0cos −=σ
d) 1516.8cot =σ e) 3424.2csc −=σ f) 0sec =σ
g) 6951.0cos =σ h) 7571.0tan −=σ i) 5.1sin =σ
j) 1tan =σ k) 1cos =σ l) 1sin =σ
5. Giventhepoint ( )yxP , lyingontheterminalarmofangleσ ,
i)statethevalueofσ ,usingbothacounter-clockwiseandaclockwiserotation
ii)determinetheprimarytrigonometricratios
a) ( )1,1 −−P b) ( )1,0 −P c) ( )0,1−P d) ( )0,1P
6. Given!"#$ = − !!, findallpossibleexactvaluesoftheothertwoprimarytrigratiosfor0 ≤ ! ≤ 360°.
7. Angle!isinthethirdquadrantand!"#$ = !!. Findallpossibleexactvaluesofthereciprocaltrigratios.
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Answers
1. a)Quadrant4,! = 45!,negative b)Quadrant2,! = 70!,negative
c)Quadrant4,! = 75!,positive d)Quadrant3,! = 45!,positive
2.
a)or 66,
511
tan,1.125
cos,1.12
11sin,1.12 ===== σσσσ
b) or 159,83
tan,5.88
cos,5.83
sin,5.8 =−=−=== σσσσ
c) or 238,58
tan,4.95
cos,4.98
sin,4.9 ==−=−== σσσσ
d) or 307,68
tan,106
cos,108
sin,10 =−==−== σσσσ
3.Answersmayvary.
4.
a)! = 29! , 151! b)! = 171! , 351! c)! = 151! , 209!d)! = 7! , 187! e)! = 205! , 335! f)Nosolution
g)! = 46! , 314! h)! = 143! , 323! i)Nosolution
j)! = 45! , 225! k)! = 0! , 360! l)! = 90!
5.
a)! = 225! ,−135! , sin! = − !! , cos! = − !
! , tan! = 1b)! = 270! ,−90! , sin! = −1, cos! = 0, tan! undefinedc)! = 180! ,−180! , sin! = 0, cos! = −1, tan! = 0d)! = 0! , sin! = 0, cos! = 1, tan! = 0
6.!"#$ = ± !! ,!"#$ = ± !
!
7.!"!# = − !!, !"#$ = − !
!,!"#$ =!!
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MCR3U–Unit5:TrigonometricRatios–Lesson3 Date:___________Learninggoal:Icandetermineandusetheexactvaluefortrigratioswhensolvingproblemswithspecialangles.
SpecialTriangles
Twospecialtrianglesweknowofareisoscelestrianglesandequilateraltriangles.Firstlet’slookatthe
primarytrigratiosinrelationtoanisoscelestriangle.
ISOCELESTRIANGLEINVESTIGATION
1. Totherightisanisoscelestrianglewithsidelengthsof1.
2. Calculate!,leaveyouranswerinexactform.
3. Findmissingangles.
4. Writethe3primarytrigratios. 5. Writethe3reciprocaltrigratios.
OneofourmaingoalsinMCR3Uistoimproveaccuracybyusingexactvalues.Fromnowonwewillalwaysusethisexactvalue.
Example1:Findexactvalueforthefollowing
a)!"# 45° b)!"#135° c)!"#315°
1
1
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EQUILATERALTRIANGLEINVESTIGATION
1. Totherightisanequilateraltrianglewithsidelengthsof2.
2. Splitthetriangleinhalfata90°angle.3. Calculatetheheightofthetriangle.
4. Findeachanglethe“split”triangle.
5. Write3primarytrigratiosfor30°.
6. Writethe3reciprocaltrigratios30°.
7. Write3primarytrigratiosfor60°.
8. Writethe3reciprocaltrigratios60°.
Again,wewillalwaysusetheseexactvalueswhenevaluatingatrigratiofor30°and60°.
Example2:Findexactvalueforthefollowing
a)sin (−210°) b)sec (−240°) c)cot (−210°)
HW:Pg.532#1-12,15(ignorequestionswithpi)
2
2
2
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MCR3U–Unit5:TrigonometricRatios–Lesson4 Date:___________Learninggoal:IcanprovetrigonometricidentitiesusingPythagoreanandquotientidentities.
TrigIdentitiesAnequationthatisalwaystrueiscalledanidentity.Wealreadyknowmanytrigidentities…
PythagoreanTheorem X-Y-RIdentities ReciprocalIdentities
!! + !! = !!
!"#$ = !!
!"#$ = !!
!"#$ = !!
!"!# = 1!"#$
!"#$ = 1!"#$
!"#$ = 1!"#$
Toprovethatagiventrigonometricequationisanidentity,bothsidesoftheequationneedtobeshowntobeequivalent(LS=RS).Thiscanbedonebysimplifyingthemorecomplicatedsideuntilitisidenticaltotheothersideormanipulatingbothsidestogetthesameexpression.
Example1:Provethefollowingidentities.
a) !"#$ = !"#$!"#$
b) !"#!! + !"#!! = 1
!"#$ = !"#$!"#$iscalled
the_____________________________.Youmaynowacceptitastrueanduseittosolveotheridentities.
!"#!! + !"#!! = 1iscalledthe_______________
______________.Youmaynowacceptitastrueanduseittosolveotheridentities.
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Example2:Provethefollowingidentities.
a)!"#$ = sin!!sin!!cos! !!sin! if !"#$ ≠ 0 b) 1+ !"#!! = !"!!!
c)cos!!!sin!!
cos!!!sin!cos! = 1− !"#$ d) tan!! − sin!! = sin!!tan!!
HW:Pg.538#1,7-9,11,13,14,16,SimplifyingExpressionsWorksheet
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SimplifyingExpressionsWorksheet
Simplify the following expressions
Algebraic Expressions Trigonometric Expressions a)
€
x + x a)
€
sinθ + sinθ
b)
€
x − y + x − y b)
€
sinθ − cosθ + sinθ − cosθ
c)
€
(x)(x) c)
€
(tanθ)(tanθ)
d)
€
(x)(y) d)
€
(sinθ)(cosθ)
e)
€
x(x +1) e)
€
tanθ(tanθ +1)
f)
€
x(2x − y) f)
€
sinθ(2sinθ − cosθ)
g)
€
(1− x)(1+ x) g)
€
(1− sinθ)(1+ sinθ)
h)
€
x − 1x
h)
€
cosθ − 1cosθ
i)
€
1x
+1y
i)
€
1sinθ
+1
cosθ
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Factor the following expressions
Algebraic Expressions Trigonometric Expressions
a)
€
x 2 + xy a)
€
sin2θ + sinθ cosθ
b)
€
x − x 2 b)
€
tanθ − tan2θ
c)
€
x 2 − 2x +1 c)
€
sin2θ − 2sinθ +1
d)
€
x 2 + 2xy+ y 2 d)
€
cos2θ + 2cosθ sinθ + sin2θ
e)
€
x 2 −1 e)
€
sin2θ −1
f)
€
1− x 2 f)
€
1− cos2θ
g)
€
x 4 −1 g)
€
tan4 θ −1
h)
€
x 4 − y 4 h)
€
sin4 θ − cos4 θ
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Understanding Trigonometric Expressions WorksheetMCR3U 4.4.1
=2 x
=2x−2 y
=x 2
=xy
=x 2+x
=2x2−xy
=1−x+x−x2
=1−x2 or
=−x2+1 or =−(x2−1) or =−( x+1) ( x−1)
=x 2−1x
=(x+1) ( x−1 )x
=y+ xxy
or
=x+ yxy
=2sin θ
=2sin θ−2cosθ=2(sin θ−cosθ)
=( tan θ )2
= tan2θ
=sin θ cosθ
= tan2 θ+ tan θ
=2sin2θ−sin θ cosθ
=1−sin θ+sin θ−sin2θ=1−sin2θ or
=−sin2θ+1 or =−(sin2θ−1) or =−(sin θ+1 ) (sin θ−1)
=cos
2θ−1cosθ
=(cosθ+1) (cosθ−1)cosθ
=cosθ+sinθsin θcosθ
or
=sin θ+cos θsin θcosθ
SOLUTIONS
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Understanding Trigonometric Expressions WorksheetMCR3U 4.4.1
=x ( x+ y )
=x (1−x)
=( x−1) ( x−1)=( x−1)2
=( x+ y) ( x+ y )=( x+ y)2
=( x+1) (x−1)
=(1+x ) (1−x ) or=( x+1) (−x+1)=−( x+1) ( x−1)
=(x 2+1)( x2−1)¿ (x2+1) ( x+1) ( x−1)
=(x 2+ y2 )(x2− y2)=( x2+ y2 )( x+ y) ( x− y )
=sin θ (sin θ+cos θ)
=tan θ (1−tan θ)
=(sin x−1) (sin x−1)=(sin x−1)2
=( cosθ+sin θ) (cos θ+sinθ )=( cosθ+sin θ)2
=(sin θ+1) (sin θ−1)
=(1+cosθ) (1−cos θ )
=( tan2 θ+1)( tan2θ−1)=( tan2θ+1) (tan θ+1) (tan θ−1)
=(sin2θ+cos2θ) (sin2θ−cos2θ)=(sin2θ+cos2θ) (sin θ+cosθ ) (sin θ−cos θ )= (sin θ+cos θ ) (sin θ−cosθ ) (enriched )
SOLUTIONS
or =−x2+x=−x ( x−1)
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MCR3U–Unit5:TrigonometricRatios–Lesson5 Date:___________Learninggoal:Icanderivethereciprocalidentitiesandusethemtoproveothertrigidentities.
ReciprocalTrigIdentitiesSUMMARY
Startwith!"#!! + !"#!! = 1thendivideby!"#!!.
Startwith!"#!! + !"#!! = 1thendivideby!"#!!.
ReciprocalIdentities QuotientIdentities PythagoreanTheorem
!"!# = 1!"#$
!"#$ = 1!"#$
!"#$ = 1!"#$
!"#$ = !"#$!"#$
!"#$ = !"#$!"#$
!"#!! + !"#!! = 1
!"#!! = 1− !"#!!
!"#!! = 1− !"#!!
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Example1:Provethefollowingtrigidentities.
a)!"#$ !"#$ − 1 = 1− !"#$ b)!"#$!"#!! − !"#$ = !"#$
HW:TrigonometricIdentitiesII&IIIWorksheet,TrigSimplificationsMatchingWorksheet
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Trigonometric Identities II WorksheetMCR3U 4.5.1
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Trigonometric Identities III Worksheet
1coscos2sin)
cossin
tan2
sin1
1
sin1
1)1cos2sincos)
tansectansec
1)
cos
tansec
sin1
1)
1csc
1csc
sin1
sin1)secsin
cot
1)
424
222
2
xxxg
xx
x
xxfxxxe
xxxx
dx
xx
xc
x
x
x
xbxx
xa
1. Prove each of the following identities:
1cot
tan1
cot1
tan1)cossin21cossin)
tan
sinsin1)tan1
cos
11
cos
1)
tansinsintan)1cot1sin)
sinsin
1
cos
1
sin
1)sincoscossin)
1seccsc
cscsec)seccsc
cossin
tancot)
tan1cossincos
sincos)
cos
1tansincos)
sin1sin1
cos)
2
2
222
222222
4222
4242
22
2222
2
22
2
x
x
x
xtxxxxs
x
xxrx
xxq
xxxxpxxo
xxxxnxxxxm
xx
xxlxx
xx
xxk
xxxx
xxj
xxxxi
xx
xh
x
xx
xx
xxzx
xxxy
xxxxxxxxx
xxw
xx
x
x
xvx
x
x
x
xu
2
2
2
22
2
22
2222
2
sin
coscos
3cos3sin
1cos2sin)cos1
cos
1costan)
tansinsintan)cossincossin
cossin21)
sec2csc1
csc
csc1
csc)sec2
sin1
cos
cos
sin1)
MCR3U 4.5.3
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MCR3U Trig Simplifications – Matching Worksheet
3sin
sin2sin392)8____
sec
csctan4csctansin)7____
cos4cossec2
cos16seccos6sec)6____
sin4cotsin
sin16sincos)5____
1sec
sectan1cos)4____
3cot2cot
9cot)3____
seccsc
cscsec)2____
cos
coscossincossin)1____
2
2
22
2
2
2
2
22
22
32
Each of the expressions on the left can be simplified and matched to an answer on
the right. After working very hard to simplify each expression and very neatly show
your rough work, place the appropriate letter the in space provided before the
expression. Letters may be used more than once. Have fun in the sandbox!
4.5.2
1sin2.
1sin
cos2sin3.
1sin.
sincos
sin3cos
1cos.
4cos2
1.
1sin.
4sin.
4cos.
sin.
1.
2
2
K
J
I
H
G
F
E
D
C
B
A