Chapter4/5Part2-TrigIdentities

andEquations

LessonPackage

MHF4U

Chapter4/5Part2OutlineUnitGoal:Bytheendofthisunit,youwillbeabletosolvetrigequationsandprovetrigidentities.

Section Subject LearningGoals CurriculumExpectations

L1 TransformationIdentities

-recognizeequivalenttrigexpressionsbyusinganglesinarighttriangleandbyperformingtransformations

B3.1

L2 CompountAngles -understanddevelopmentofcompoundangleformulasandusethemtofindexactexpressionsfornon-specialangles B3.2

L3 DoubleAngle -usecompoundangleformulastoderivedoubleangleformulas-useformulastosimplifyexpressions B3.3

L4 ProvingTrigIdentities -Beabletoproveidentitiesusingidentitieslearnedthroughouttheunit B3.3

L5 SolveLinearTrigEquations

-Findallsolutionstoalineartrigequation B3.4

L5 SolveTrigEquationswithDoubleAngles

-Findallsolutionstoatrigequationinvolvingadoubleangle B3.4

L5 SolveQuadraticTrigEquations

-Findallsolutionstoaquadratictrigequation B3.4

L5 ApplicationsofTrigEquations

-Solveproblemsarisingfromrealworldapplicationsinvolvingtrigequations B2.7

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P Quiz–SolvingTrigEquations F P PreTestReview F/A P Test–TrigIdentitiesandEquations O B3.1,3.2,3.3,3.4

B2.7 P K(21%),T(34%),A(10%),C(34%)

L1–4.3Co-functionIdentitiesMHF4UJensenPart1:RememberingHowtoProveTrigIdentities

TipsandTricksReciprocalIdentities QuotientIdentities PythagoreanIdentities

Squarebothsides

𝑐𝑠𝑐#𝜃 = '

()*+,

𝑠𝑒𝑐#𝜃 =1

𝑐𝑜𝑠#𝜃

𝑐𝑜𝑡#𝜃 =1

𝑡𝑎𝑛#𝜃

Squarebothsides

𝑠𝑖𝑛#𝜃𝑐𝑜𝑠#𝜃

= 𝑡𝑎𝑛#𝜃

cos# 𝜃 sin# 𝜃

= cot# 𝜃

Rearrangetheidentity

𝑠𝑖𝑛#𝜃 = 1 − 𝑐𝑜𝑠#𝜃

𝑐𝑜𝑠#𝜃 = 1 − 𝑠𝑖𝑛#𝜃

Dividebyeithersinorcos

1 + 𝑐𝑜𝑡#𝜃 = csc 𝜃

𝑡𝑎𝑛#𝜃 + 1 = 𝑠𝑒𝑐#𝜃

Generaltipsforprovingidentities:

i) Separate into LS and RS. Terms may NOT cross between sides. ii) Try to change everything to 𝑠𝑖𝑛𝜃 or 𝑐𝑜𝑠𝜃 iii) If you have two fractions being added or subtracted, find a common denominator and

combine the fractions. iv) Use difference of squares à 1 − 𝑠𝑖𝑛#𝜃 = (1 − 𝑠𝑖𝑛𝜃)(1 + 𝑠𝑖𝑛𝜃) v) Use the power rule à 𝑠𝑖𝑛>𝜃 = (𝑠𝑖𝑛#𝜃)?

FundamentalTrigonometricIdentitiesReciprocalIdentities QuotientIdentities PythagoreanIdentities

𝒄𝒔𝒄𝜽 =𝟏

𝒔𝒊𝒏𝜽

𝒔𝒆𝒄𝜽 =𝟏

𝒄𝒐𝒔𝜽

𝒄𝒐𝒕𝜽 =𝟏

𝒕𝒂𝒏𝜽

𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽

= 𝒕𝒂𝒏𝜽

𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝜽

= 𝐜𝐨𝐭 𝜽

𝒔𝒊𝒏𝟐𝜽

+

𝒄𝒐𝒔𝟐𝜽

=

𝟏

LS RS

LS=RS

LS RS

LS=RS

LS RS

LS=RS

Example1:Proveeachofthefollowingidentitiesa)tan# 𝑥 + 1 = sec# 𝑥b)cos# 𝑥 = (1 − sin 𝑥)(1 + sin 𝑥)c) TUV

+ W'XYZT W

= 1 + cos 𝑥

Part2:TransformationIdentitiesBecauseoftheirperiodicnature,therearemanyequivalenttrigonometricexpressions.Horizontaltranslationsof[

#thatinvolvebothasinefunctionandacosinefunctioncanbeusedtoobtaintwo

equivalentfunctionswiththesamegraph.Translatingthecosinefunction[

#totheright,𝑓 𝑥 = cos 𝑥 − [

#

resultsinthegraphofthesinefunction,𝑓 𝑥 = sin 𝑥.Similarly,translatingthesinefunction[

#totheleft,𝑓 𝑥 = sin 𝑥 + [

#

resultsinthegraphofthecosinefunction,𝑓 𝑥 = cos 𝑥.

Part3:Even/OddFunctionIdentitiesRememberthat𝐜𝐨𝐬 𝒙isanevenfunction.Reflectingitsgraphacrossthe𝑦-axisresultsintwoequivalentfunctionswiththesamegraph.sin 𝑥andtan 𝑥arebothoddfunctions.Theyhaverotationalsymmetryabouttheorigin.

TransformationIdentities cos `𝑥 − [

#a = sin 𝑥 sin `𝑥 + [

#a = cos𝑥

Even/OddIdentities cos(−𝑥) = cos𝑥 sin(−𝑥) = −sin 𝑥 tan(−𝑥) = − tan 𝑥

LS RS

LS=RS

Part4:Co-functionIdentitiesTheco-functionidentitiesdescribetrigonometricrelationshipsbetweencomplementaryanglesinarighttriangle.

WecouldidentifyotherequivalenttrigonometricexpressionsbycomparingprincipleanglesdrawninstandardpositioninquadrantsII,III,andIVwiththeirrelatedacute(reference)angleinquadrantI.

PrincipleinQuadrantII PrincipleinQuadrantIII PrincipleinQuadrantIV

sin(𝜋 − 𝑥) = sin 𝑥 sin(𝜋 + 𝑥) = −sin 𝑥 sin(2𝜋 − 𝑥) = −sin 𝑥Example2:Provebothco-functionidentitiesusingtransformationidentitiesa)cos [

#− 𝑥 = sin 𝑥 b)sin [

#− 𝑥 = cos 𝑥

Co-FunctionIdentities cos `[

#− 𝑥a = sin 𝑥 sin `[

#− 𝑥a = cos𝑥

Part5:ApplytheIdentitiesExample3:Giventhatsin [

d≅ 0.5878,useequivalenttrigonometricexpressionstoevaluatethefollowing:

b)cos j['k

= sin l𝜋2 −

7𝜋10m

= sin l5𝜋10 −

7𝜋10m

= sin l−2𝜋10m

= −sin `𝜋5a

≅ −0.5878

a)cos ?['k

= sin l𝜋2 −

3𝜋10m

= sin l5𝜋10 −

3𝜋10m

= sin l2𝜋10m

= sin `𝜋5a

≅ 0.5878

L2–4.4CompoundAngleFormulasMHF4UJensenCompoundangle:ananglethatiscreatedbyaddingorsubtractingtwoormoreangles.Part1:Proofof𝐜𝐨𝐬(𝒙 − 𝒚)Normalalgebrarulesdonotapply:

cos(𝑥 − 𝑦) ≠ cos 𝑥 − cos 𝑦Sowhatdoescos 𝑥 − 𝑦 =?Considerthediagramtotheright…Bythecosinelaw:𝑐2 = 12 + 12 − 2 1 1 cos(𝑎 − 𝑏)𝑐2 = 2 − 2cos(𝑎 − 𝑏)THISISEQUATION1Butnoticethat𝑐hasendpointsof(cos 𝑎 , sin 𝑎)and(cos 𝑏 , sin 𝑏)Usingthedistanceformula𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑥2 − 𝑥B 2 + 𝑦2 − 𝑦B 2𝑐 = cos 𝑎 − cos 𝑏 2 + sin 𝑎 − sin 𝑏 2𝑐2 = cos 𝑎 − cos 𝑏 2 + sin 𝑎 − sin 𝑏 2𝑐2 = cos2 𝑎 − 2 cos 𝑎 cos 𝑏 + cos2 𝑏 + sin2 𝑎 − 2 sin 𝑎 sin 𝑏 + sin2 𝑏𝑐2 = 1 − 2 cos 𝑎 cos 𝑏 − 2 sin 𝑎 sin 𝑏 + 1𝑐2 = 2 − 2 cos 𝑎 cos 𝑏 − 2 sin 𝑎 sin 𝑏THISISEQUATION2Setequations1and2equal2 − 2 cos 𝑎 − 𝑏 = 2 − 2 cos 𝑎 cos 𝑏 − 2 sin 𝑎 sin 𝑏−2 cos 𝑎 − 𝑏 = −2 cos 𝑎 cos 𝑏 − 2 sin 𝑎 sin 𝑏

𝐜𝐨𝐬 𝒂 − 𝒃 = 𝐜𝐨𝐬𝒂 𝐜𝐨𝐬 𝒃 + 𝐬𝐢𝐧 𝒂 𝐬𝐢𝐧 𝒃

Part2:ProofsofothercompoundangleformulasExample1:Provecos 𝑥 + 𝑦 = cos 𝑥 cos 𝑦 − sin 𝑥 sin 𝑦

Example2:Proveadditionandsubtractionformulasforsineusingco-functionidentitiesandthesubtractionformulaforcosine.a)Provesin 𝑥 + 𝑦 = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

LS= cos(𝑥 + 𝑦)= cos[𝑥 − (−𝑦)]= cos 𝑥 cos(−𝑦) + sin 𝑥 sin(−𝑦)= cos 𝑥 cos 𝑦 + sin 𝑥 (−sin 𝑦)= cos 𝑥 cos 𝑦 − sin 𝑥 sin 𝑦

RS= cos 𝑥 cos 𝑦 − sin 𝑥 sin 𝑦

LS=RS

Co-FunctionIdentitiescos IJ

2− 𝑥K = sin 𝑥 sin IJ

2− 𝑥K = cos 𝑥

LS= sin(𝑥 + 𝑦)= cos L

𝜋

2 − (𝑥 + 𝑦)N

= cos LI

𝜋

2 − 𝑥K− 𝑦N

= cos I

𝜋

2 − 𝑥K cos 𝑦 + sin I𝜋

2 − 𝑥K (sin 𝑦)= sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

RS= sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

LS=RS

b)Provesin 𝑥 − 𝑦 = sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦

LS= sin(𝑥 − 𝑦)= sin[𝑥 + (−𝑦)]= sin 𝑥 cos(−𝑦) + cos𝑥 sin(−𝑦)= sin 𝑥 cos 𝑦 + cos 𝑥 (−sin 𝑦)= sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦

RS= sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦

LS=RS

CompoundAngleFormulas

sin(𝑥 + 𝑦) = sin 𝑥 cos𝑦 + cos 𝑥 sin 𝑦

sin(𝑥 − 𝑦) = sin 𝑥 cos𝑦 − cos 𝑥 sin 𝑦

cos(𝑥 + 𝑦) = cos𝑥 cos𝑦 − sin 𝑥 sin 𝑦

cos(𝑥 − 𝑦) = cos𝑥 cos𝑦 + sin 𝑥 sin 𝑦

tan(𝑥 + 𝑦) =tan 𝑥 + tan 𝑦1 − tan 𝑥 tan 𝑦

tan(𝑥 − 𝑦) =tan 𝑥 − tan 𝑦1 + tan 𝑥 tan 𝑦

1

1√2

2

1

√3

Part3:DetermineExactTrigRatiosforAnglesotherthanSpecialAnglesByexpressinganangleasasumordifferenceofanglesinthespecialtriangles,exactvaluesofotheranglescanbedetermined.Example3:Usecompoundangleformulastodetermineexactvaluesfor

b)tan I−QJB2K

tan R−5𝜋12T = − tan R

5𝜋12T

= − tan R2𝜋12 +

3𝜋12T

= −tan I2𝜋12K + tan I

3𝜋12K

1 − tan I2𝜋12K tan I3𝜋12K

= −tan I𝜋6K + tan I

𝜋4K

1 − tan I𝜋6K tan I𝜋4K

= −

1√3

+ 1

1 − 1√3

= −

1√3

+ √3√3

√3√3

− 1√3

= −

1 + √3√3

√3 − 1√3

= −1 + √3√3 − 1

a)sin JB2

sin𝜋12 = sin R

4𝜋12 −

3𝜋12T

= sin IJ

Y− J

ZK

= sin IJ

YK cos IJ

ZK − cos IJ

YK sin IJ

ZK

= √Y2I B√2K − IB

2K I B

√2K

= √Y2√2

− B2√2

= √Y[B2√2

Part4:UseCompoundAngleFormulastoSimplifyTrigExpressionsExample4:Simplifythefollowingexpression

cos7𝜋12 cos

5𝜋12 + sin

7𝜋12 sin

5𝜋12

= cos7𝜋12 −

5𝜋12

= cos2𝜋12

= cos𝜋6

=32

Part5:ApplicationExample5:Evaluatesin(𝑎 + 𝑏),where𝑎and𝑏arebothanglesinthesecondquadrant;givensin 𝑎 = Y

Qand

sin 𝑏 = QBY

Startbydrawingbothterminalarmsinthesecondquadrantandsolvingforthethirdside.sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏

=35 −

1213 + −

45

513

= −3665 −

2065

= −5665

L3–4.5DoubleAngleFormulasMHF4UJensenPart1:ProofsofDoubleAngleFormulasExample1:Provesin 2𝑥 = 2 sin 𝑥 cos 𝑥

Example2:Provecos 2𝑥 = cos) 𝑥 − sin) 𝑥

Note:Therearealternateversionsof𝑐𝑜𝑠 2𝑥whereeither𝑐𝑜𝑠) 𝑥 𝑂𝑅 𝑠𝑖𝑛) 𝑥arechangedusingthePythagoreanIdentity.

LS= sin(2𝑥)= sin(𝑥 + 𝑥)= sin 𝑥 cos 𝑥 + cos 𝑥 sin 𝑥= 2 sin 𝑥 cos 𝑥

RS= 2 sin 𝑥 cos 𝑥

LS=RS

LS= cos(2𝑥)= cos(𝑥 + 𝑥)= cos 𝑥 cos 𝑥 − sin 𝑥 sin 𝑥= cos) 𝑥 − sin) 𝑥

RS= cos) 𝑥 − sin) 𝑥

LS=RS

1

1√2

2

1

√3

Part2:UseDoubleAngleFormulastoSimplifyExpressionsExample1:Simplifyeachofthefollowingexpressionsandthenevaluatea)2 sin 6

7cos 6

7

= sin 2𝜋8

= sin𝜋4

=12

b)) <=>?@AB<=>C?@

= tan 2𝜋6

= tan𝜋3

= 3

DoubleAngleFormulas

sin(2𝑥) = 2 sin 𝑥 cos 𝑥

cos(2𝑥) = cos) 𝑥 − sin) 𝑥

cos(2𝑥) = 2 cos) 𝑥 − 1

cos(2𝑥) = 1 − 2 sin) 𝑥

tan(2𝑥) =2 tan 𝑥

1 − tan) 𝑥

Part3:DeterminetheValueofTrigRatiosforaDoubleAngleIfyouknowoneoftheprimarytrigratiosforanyangle,thenyoucandeterminetheothertwo.Youcanthendeterminetheprimarytrigratiosforthisangledoubled.

Example2:Ifcos 𝜃 = − )Iand0 ≤ 𝜃 ≤ 2𝜋,determinethevalueofcos(2𝜃)andsin(2𝜃)

Wecansolveforcos 2𝜃 withoutfindingthesineratioifweusethefollowingversionofthedoubleangleformula:cos 2𝜃 = 2 cos) 𝜃 − 1

cos 2𝜃 = 2 −23

)

− 1

cos 2𝜃 = 249 − 1

cos 2𝜃 =89 −

99

cos 2𝜃 = −19

Tofindsin(2𝜃)wewillneedtofindsin 𝜃usingthecosineratiogiveninthequestion.Sincetheoriginalcosineratioisnegative,𝜃couldbeinquadrant2or3.Wewillhavetoconsiderbothscenarios.

Scenario1:𝜃inQuadrant2 Scenario2:𝜃inQuadrant3

cos 2𝜃 = − AMandsin 2𝜃 = − N O

MorN O

M

Example3:Iftan 𝜃 = − INandI6

)≤ 𝜃 ≤ 2𝜋,determinethevalueofcos(2𝜃).

Wearegiventhattheterminalarmoftheangleliesinquadrant4:

LS RS

L4–4.5ProveTrigIdentitiesMHF4UJensenUsingyoursheetofallidentitieslearnedthisunit,proveeachofthefollowing:Example1:Prove !"#(%&)

()*+!(%&)= tan 𝑥

Example2:Provecos 4

%+ 𝑥 = −sin 𝑥

LS RS

Example3:Provecsc(2𝑥) = *!* &% *+! &

Example4:Provecos 𝑥 = (

*+! &− sin 𝑥 tan 𝑥

Example5:Provetan(2𝑥) − 2 tan 2𝑥 sin% 𝑥 = sin 2𝑥Example6:Prove*+!(&9:)

*+!(&):)= ();<#& ;<#:

(9;<# & ;<#:

L5–5.4SolveLinearTrigonometricEquationsMHF4UJensenInthepreviouslessonwehavebeenworkingwithidentities.IdentitiesareequationsthataretrueforANYvalueof𝑥.Inthislesson,wewillbeworkingwithequationsthatarenotidentities.Wewillhavetosolveforthevalue(s)thatmaketheequationtrue.Rememberthat2solutionsarepossibleforananglebetween0and2𝜋withagivenratio.UsethereferenceangleandCASTruletodeterminetheangles.Whensolvingatrigonometricequation,considerall3toolsthatcanbeuseful:

1. SpecialTriangles2. GraphsofTrigFunctions3. Calculator

Example1:Findallsolutionsforcos 𝜃 = − *+intheinterval0 ≤ 𝑥 ≤ 2𝜋

Example2:Findallsolutionsfortan 𝜃 = 5intheinterval0 ≤ 𝑥 ≤ 2𝜋Example3:Findallsolutionsfor2 sin 𝑥 + 1 = 0intheinterval0 ≤ 𝑥 ≤ 2𝜋

Example4:Solve3 tan 𝑥 + 1 = 2,where0 ≤ 𝑥 ≤ 2𝜋

L6–5.4SolveDoubleAngleTrigonometricEquationsMHF4UJensenPart1:Investigation 𝒚 = 𝐬𝐢𝐧𝒙 𝒚 = 𝐬𝐢𝐧(𝟐𝒙)a)Whatistheperiodofbothofthefunctionsabove?Howmanycyclesbetween0and2𝜋radians?For𝑦 = sin 𝑥à𝑝𝑒𝑟𝑖𝑜𝑑 = 2𝜋For𝑦 = sin(2𝑥)à𝑝𝑒𝑟𝑖𝑜𝑑 = 89

8= 𝜋

b)Lookingatthegraphof𝑦 = sin 𝑥,howmanysolutionsarethereforsin 𝑥 = :

8≈ 0.71?

2solutions

sin𝜋4 = sin

3𝜋4 =

12

c)Lookingatthegraphof𝑦 = sin(2𝑥),howmanysolutionsarethereforsin(2𝑥) = :

8≈ 0.71?

4solutions

sin𝜋8 = sin

3𝜋8 = sin

9𝜋8 = sin

11𝜋8 =

12

d)Whentheperiodofafunctioniscutinhalf,whatdoesthatdotothenumberofsolutionsbetween0and2𝜋radians?Doublesthenumberofsolutions

Part2:SolveLinearTrigonometricEquationsthatInvolveDoubleAngles

Example1:sin(2𝜃) = D8where0 ≤ 𝜃 ≤ 2𝜋

Example2:cos(2𝜃) = − :8where0 ≤ 𝜃 ≤ 2𝜋

Example3:tan(2𝜃) = 1where0 ≤ 𝜃 ≤ 2𝜋

L7–5.4SolveQuadraticTrigonometricEquationsMHF4UJensenAquadratictrigonometricequationmayhavemultiplesolutionsintheinterval0 ≤ 𝑥 ≤ 2𝜋.Youcanoftenfactoraquadratictrigonometricequationandthensolvetheresultingtwolineartrigonometricequations.Incaseswheretheequationcannotbefactored,usethequadraticformulaandthensolvetheresultinglineartrigonometricequations.YoumayneedtouseaPythagoreanidentity,compoundangleformula,ordoubleangleformulatocreateaquadraticequationthatcontainsonlyasingletrigonometricfunctionwhoseargumentsallmatch.Rememberthatwhensolvingalineartrigonometricequation,considerall3toolsthatcanbeuseful:

1. SpecialTriangles2. GraphsofTrigFunctions3. Calculator

Part1:SolvingQuadraticTrigonometricEquations

Example1:Solveeachofthefollowingequationsfor0 ≤ 𝑥 ≤ 2𝜋

a) sin 𝑥 + 1 sin 𝑥 − ,-= 0

b)sin- 𝑥 − sin 𝑥 = 2

c)2sin- 𝑥 − 3 sin 𝑥 + 1 = 0

Part2:UseIdentitiestoHelpSolveQuadraticTrigonometricEquationsExample2:Solveeachofthefollowingequationsfor0 ≤ 𝑥 ≤ 2𝜋a)2sec- 𝑥 − 3 + tan 𝑥 = 0

b)3 sin 𝑥 + 3 cos(2𝑥) = 2

L8–5.4ApplicationsofTrigonometricEquationsMHF4UJensenPart1:ApplicationQuestions

Example1:Today,thehightideinMatthewsCove,NewBrunswick,isatmidnight.Thewaterlevelathightideis7.5m.Thedepth,dmeters,ofthewaterinthecoveattimethoursismodelledbytheequation

𝑑 𝑡 = 3.5 cos *+𝑡 + 4

Jennyisplanningadaytriptothecovetomorrow,butthewaterneedstobeatleast2mdeepforhertomaneuverhersailboatsafely.DeterminethebesttimewhenitwillbesafeforhertosailintoMatthewsCove?

Example2:Acity’sdailytemperature,indegreesCelsius,canbemodelledbythefunction

𝑡 𝑑 = −28 cos 1*2+3

𝑑 + 10

where𝑑isthedayoftheyearand1=January1.Ondayswherethetemperatureisapproximately32°Corabove,theairconditionersatcityhallareturnedon.Duringwhatdaysoftheyeararetheairconditionersrunningatcityhall?

Example3:AFerriswheelwitha20meterdiameterturnsonceeveryminute.Ridersmustclimbup1metertogetontheride.a)Writeacosineequationtomodeltheheightoftherider,ℎmeters,𝑡secondsaftertheridehasbegun.Assumetheystartattheminheight.

𝑎 =𝑚𝑎𝑥 −𝑚𝑖𝑛

2 =21 − 12 = 10

𝑘 = 1𝜋

?@ABCD= 1𝜋

+E= 𝜋

2E

𝑐 = 𝑚𝑎𝑥 − 𝑎 = 21 − 10 = 11𝑑GCH = 30ℎ(𝑡) = 10 cos

𝜋30 𝑡 − 30 + 11

b)Whatwillbethefirst2timesthattheriderisataheightof5meters?