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