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.

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

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!"#$

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)

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

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.

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.

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

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.

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.!"!# = − !!, !"#$ = − !

!,!"#$ =!!

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

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

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.

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

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

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

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

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)

MCR3U–Unit5:TrigonometricRatios–Lesson5 Date:___________Learninggoal:Icanderivethereciprocalidentitiesandusethemtoproveothertrigidentities.

ReciprocalTrigIdentitiesSUMMARY

Startwith!"#!! + !"#!! = 1thendivideby!"#!!.

Startwith!"#!! + !"#!! = 1thendivideby!"#!!.

ReciprocalIdentities QuotientIdentities PythagoreanTheorem

!"!# = 1!"#$

!"#$ = 1!"#$

!"#$ = 1!"#$

!"#$ = !"#$!"#$

!"#$ = !"#$!"#$

!"#!! + !"#!! = 1

!"#!! = 1− !"#!!

!"#!! = 1− !"#!!

Example1:Provethefollowingtrigidentities.

a)!"#$ !"#$ − 1 = 1− !"#$ b)!"#$!"#!! − !"#$ = !"#$

HW:TrigonometricIdentitiesII&IIIWorksheet,TrigSimplificationsMatchingWorksheet

Trigonometric Identities II WorksheetMCR3U 4.5.1

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

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