chapter 4/5 part 2- trig identities and equations unit 5 lesson package teacher.pdf · chapter 4/5...
TRANSCRIPT
![Page 1: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/1.jpg)
Chapter4/5Part2-TrigIdentities
andEquations
LessonPackage
MHF4U
![Page 2: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/2.jpg)
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%)
![Page 3: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/3.jpg)
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
![Page 4: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/5.jpg)
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
![Page 6: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/7.jpg)
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
![Page 8: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/8.jpg)
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
![Page 9: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/9.jpg)
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
![Page 10: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/10.jpg)
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
![Page 11: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/11.jpg)
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
![Page 12: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/12.jpg)
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
![Page 13: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/13.jpg)
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
![Page 14: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/14.jpg)
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
![Page 15: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/15.jpg)
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
![Page 16: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/16.jpg)
Example3:Iftan 𝜃 = − INandI6
)≤ 𝜃 ≤ 2𝜋,determinethevalueofcos(2𝜃).
Wearegiventhattheterminalarmoftheangleliesinquadrant4:
![Page 17: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/17.jpg)
LS RS
L4–4.5ProveTrigIdentitiesMHF4UJensenUsingyoursheetofallidentitieslearnedthisunit,proveeachofthefollowing:Example1:Prove !"#(%&)
()*+!(%&)= tan 𝑥
Example2:Provecos 4
%+ 𝑥 = −sin 𝑥
![Page 18: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/18.jpg)
LS RS
Example3:Provecsc(2𝑥) = *!* &% *+! &
Example4:Provecos 𝑥 = (
*+! &− sin 𝑥 tan 𝑥
![Page 19: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/19.jpg)
Example5:Provetan(2𝑥) − 2 tan 2𝑥 sin% 𝑥 = sin 2𝑥Example6:Prove*+!(&9:)
*+!(&):)= ();<#& ;<#:
(9;<# & ;<#:
![Page 20: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/20.jpg)
L5–5.4SolveLinearTrigonometricEquationsMHF4UJensenInthepreviouslessonwehavebeenworkingwithidentities.IdentitiesareequationsthataretrueforANYvalueof𝑥.Inthislesson,wewillbeworkingwithequationsthatarenotidentities.Wewillhavetosolveforthevalue(s)thatmaketheequationtrue.Rememberthat2solutionsarepossibleforananglebetween0and2𝜋withagivenratio.UsethereferenceangleandCASTruletodeterminetheangles.Whensolvingatrigonometricequation,considerall3toolsthatcanbeuseful:
1. SpecialTriangles2. GraphsofTrigFunctions3. Calculator
Example1:Findallsolutionsforcos 𝜃 = − *+intheinterval0 ≤ 𝑥 ≤ 2𝜋
![Page 21: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/21.jpg)
Example2:Findallsolutionsfortan 𝜃 = 5intheinterval0 ≤ 𝑥 ≤ 2𝜋Example3:Findallsolutionsfor2 sin 𝑥 + 1 = 0intheinterval0 ≤ 𝑥 ≤ 2𝜋
![Page 22: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/22.jpg)
Example4:Solve3 tan 𝑥 + 1 = 2,where0 ≤ 𝑥 ≤ 2𝜋
![Page 23: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/23.jpg)
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
![Page 24: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/24.jpg)
Part2:SolveLinearTrigonometricEquationsthatInvolveDoubleAngles
Example1:sin(2𝜃) = D8where0 ≤ 𝜃 ≤ 2𝜋
![Page 25: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/25.jpg)
Example2:cos(2𝜃) = − :8where0 ≤ 𝜃 ≤ 2𝜋
![Page 26: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/26.jpg)
Example3:tan(2𝜃) = 1where0 ≤ 𝜃 ≤ 2𝜋
![Page 27: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/27.jpg)
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
![Page 28: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/28.jpg)
b)sin- 𝑥 − sin 𝑥 = 2
![Page 29: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/29.jpg)
c)2sin- 𝑥 − 3 sin 𝑥 + 1 = 0
![Page 30: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/30.jpg)
Part2:UseIdentitiestoHelpSolveQuadraticTrigonometricEquationsExample2:Solveeachofthefollowingequationsfor0 ≤ 𝑥 ≤ 2𝜋a)2sec- 𝑥 − 3 + tan 𝑥 = 0
![Page 31: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/31.jpg)
b)3 sin 𝑥 + 3 cos(2𝑥) = 2
![Page 32: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/32.jpg)
L8–5.4ApplicationsofTrigonometricEquationsMHF4UJensenPart1:ApplicationQuestions
Example1:Today,thehightideinMatthewsCove,NewBrunswick,isatmidnight.Thewaterlevelathightideis7.5m.Thedepth,dmeters,ofthewaterinthecoveattimethoursismodelledbytheequation
𝑑 𝑡 = 3.5 cos *+𝑡 + 4
Jennyisplanningadaytriptothecovetomorrow,butthewaterneedstobeatleast2mdeepforhertomaneuverhersailboatsafely.DeterminethebesttimewhenitwillbesafeforhertosailintoMatthewsCove?
![Page 33: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/33.jpg)
Example2:Acity’sdailytemperature,indegreesCelsius,canbemodelledbythefunction
𝑡 𝑑 = −28 cos 1*2+3
𝑑 + 10
where𝑑isthedayoftheyearand1=January1.Ondayswherethetemperatureisapproximately32°Corabove,theairconditionersatcityhallareturnedon.Duringwhatdaysoftheyeararetheairconditionersrunningatcityhall?
![Page 34: Chapter 4/5 Part 2- Trig Identities and Equations unit 5 lesson package TEACHER.pdf · Chapter 4/5 Part 2 Outline Unit Goal: ... - Find all solutions to a quadratic trig equation](https://reader034.vdocuments.site/reader034/viewer/2022051509/5b0e58cd7f8b9a2c3b8e8f96/html5/thumbnails/34.jpg)
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?