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

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_____thatinvolvebothasinefunctionandacosinefunctioncanbeusedtoobtaintwoequivalentfunctionswiththesamegraph.Translatingthecosinefunction[

#tothe______________________________

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

#tothe________________________

resultsinthegraphofthecosinefunction,𝑓 𝑥 = cos 𝑥.

Part3:Even/OddFunctionIdentitiesRememberthat𝐜𝐨𝐬 𝒙isan__________function.Reflectingitsgraphacrossthe𝑦-axisresultsintwoequivalentfunctionswiththesamegraph.sin 𝑥andtan 𝑥areboth__________functions.Theyhaverotationalsymmetryabouttheorigin.

TransformationIdentities

Even/OddIdentities

Part4:Co-functionIdentitiesTheco-functionidentitiesdescribetrigonometricrelationshipsbetweencomplementaryanglesinarighttriangle.

WecouldidentifyotherequivalenttrigonometricexpressionsbycomparingprincipleanglesdrawninstandardpositioninquadrantsII,III,andIVwiththeirrelatedacute(reference)angleinquadrantI.

PrincipleinQuadrantII PrincipleinQuadrantIII PrincipleinQuadrantIV

Example2:Provebothco-functionidentitiesusingtransformationidentitiesa)cos [

#− 𝑥 = sin 𝑥 b)sin [

#− 𝑥 = cos 𝑥

Co-FunctionIdentities

Part5:ApplytheIdentitiesExample3:Giventhatsin [

`≅ 0.5878,useequivalenttrigonometricexpressionstoevaluatethefollowing:

b)cos f['g

a)cos ?['g

1

x

y

L2 – 4.4 Compound Angle Formulas MHF4U Jensen

Compound angle: an angle that is created by adding or subtracting two or more angles. Normal algebra rules do not apply:

cos(𝑥 + 𝑦) ≠ cos 𝑥 + cos 𝑦 Part 1: Proof of 𝐜𝐨𝐬(𝒙 + 𝒚) and 𝐬𝐢𝐧(𝒙 + 𝒚) So what does cos(𝑥 + 𝑦) =? Using the diagram below, label all angles and sides:

cos(𝑥 + 𝑦) = sin(𝑥 + 𝑦) =

Part 2: Proofs of other compound angle formulas Example 1: Prove cos(𝑥 − 𝑦) = cos 𝑥 cos 𝑦 + sin 𝑥 sin 𝑦

Example 2: a) Prove sin(𝑥 − 𝑦) = sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦

LS

RS

LS = RS

Even/Odd Properties cos(−𝑥) = cos 𝑥 sin(−𝑥) = −sin 𝑥

LS

RS

LS = RS

1

1√2

2

1

√3

Part 3: Determine Exact Trig Ratios for Angles other than Special Angles By expressing an angle as a sum or difference of angles in the special triangles, exact values of other angles can be determined. Example 3: Use compound angle formulas to determine exact values for

Compound Angle Formulas

sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

sin(𝑥 − 𝑦) = sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦

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

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

tan(𝑥 + 𝑦) =tan 𝑥 + tan 𝑦

1 − tan 𝑥 tan 𝑦

tan(𝑥 − 𝑦) =tan 𝑥 − tan 𝑦

1 + tan 𝑥 tan 𝑦

a) sin𝜋

12

sin𝜋

12=

b) tan (−5𝜋

12)

tan (−5𝜋

12) =

Part 4: Use Compound Angle Formulas to Simplify Trig Expressions Example 4: Simplify the following expression

cos7𝜋

12cos

5𝜋

12+ sin

7𝜋

12sin

5𝜋

12

Part 5: Application

Example 5: Evaluate sin(𝑎 + 𝑏), where 𝑎 and 𝑏 are both angles in the second quadrant; given sin 𝑎 =3

5 and

sin 𝑏 =5

13

Start by drawing both terminal arms in the second quadrant and solving for the third side.

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

RS

LS

RS

Part2:UseDoubleAngleFormulastoSimplifyExpressionsExample1:Simplifyeachofthefollowingexpressionsandthenevaluatea)2 sin 3

4cos 3

4

b)) 56789:;567<89

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 = − )Cand0 ≤ 𝜃 ≤ 2𝜋,determinethevalueofcos(2𝜃)andsin(2𝜃)

Wecansolveforcos 2𝜃 withoutfindingthesineratioifweusethefollowingversionofthedoubleangleformula:Tofindsin(2𝜃)wewillneedtofindsin 𝜃usingthecosineratiogiveninthequestion.Sincetheoriginalcosineratioisnegative,𝜃couldbeinquadrant_________.Wewillhavetoconsiderbothscenarios.

Scenario1:𝜃inQuadrant___ Scenario2:𝜃inQuadrant___

Example3:Iftan 𝜃 = − CGandC3

)≤ 𝜃 ≤ 2𝜋,determinethevalueofcos(2𝜃).

Wearegiventhattheterminalarmoftheangleliesinquadrant___:

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

()*+!(%&)= tan 𝑥

Example2:Provecos 4

%+ 𝑥 = −sin 𝑥

LS

RS

LS

RS

Example3:Provecsc(2𝑥) = *!* &% *+! &

Example4:Provecos 𝑥 = (

*+! &− sin 𝑥 tan 𝑥

LS

RS

LS

RS

Example5:Provetan(2𝑥) − 2 tan 2𝑥 sin% 𝑥 = sin 2𝑥

Example6:Prove*+!(&9:)

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

(9;<# & ;<#:

LS

RS

LS

RS

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?b)Lookingatthegraphof𝑦 = sin 𝑥,howmanysolutionsarethereforsin 𝑥 = 2

3≈ 0.71?

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

3≈ 0.71?

d)Whentheperiodofafunctioniscutinhalf,whatdoesthatdotothenumberofsolutionsbetween0and2𝜋radians?

Part2:SolveLinearTrigonometricEquationsthatInvolveDoubleAngles

Example1:sin(2𝜃) = 93where0 ≤ 𝜃 ≤ 2𝜋

Example2:cos(2𝜃) = − 23where0 ≤ 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.b)Whatwillbethefirst2timesthattheriderisataheightof5meters?