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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 QuizβSolvingTrigEquations F P PreTestReview F/A P TestβTrigIdentitiesandEquations O B3.1,3.2,3.3,3.4
B2.7 P K(21%),T(34%),A(10%),C(34%)
L1 β 4.3 Co-function Identities MHF4U Jensen
Part 1: Remembering How to Prove Trig Identities
Tips and Tricks
Reciprocal Identities Quotient Identities Pythagorean Identities
Square both sides
ππ π2 π =1
π ππ2 π
π ππ2 π =1
πππ 2 π
πππ‘2 π =1
π‘ππ2 π
Square both sides
π ππ2 π
πππ 2 π= π‘ππ2 π
cos2 π
sin2 π = cot2 π
Rearrange the identity
π ππ2π = 1 β πππ 2π
πππ 2π = 1 β π ππ2π
Divide by either sin2 π or sin2 π
1 + πππ‘2π = csc2 π
π‘ππ2π + 1 = π ππ2π
General tips for proving identities:
i) Separate into LS and RS. Terms may NOT cross between sides.
ii) Try to change everything to π ππ π or πππ π
iii) If you have two fractions being added or subtracted, find a common denominator and
combine the fractions.
iv) Use difference of squares β 1 β π ππ2 π = (1 β π ππ π)(1 + π ππ π)
v) Use the power rule β π ππ 6 π = (π ππ2 π)3
Fundamental Trigonometric Identities Reciprocal Identities Quotient Identities Pythagorean Identities
πππ π½ =π
πππ π½
πππ π½ =π
πππ π½
πππ π½ =π
πππ π½
πππ π½
πππ π½= πππ π½
ππ¨π¬ π½
π¬π’π§ π½ = ππ¨π π½
π¬π’π§π π½ + ππ¨π¬π π½ = π
Example 1: Prove each of the following identities a) tan2 π₯ + 1 = sec2 π₯ b) cos2 π₯ = (1 β sin π₯)(1 + sin π₯)
c) sin2 π₯
1βcos π₯= 1 + cos π₯
Part 2: Transformation Identities Because of their periodic nature, there are many equivalent trigonometric expressions . Horizontal translations of _____ that involve both a sine function and a cosine function can be used to obtain two equivalent functions with the same graph.
Translating the cosine function π
2 to the ______________________________
results in the graph of the sine function, π(π₯) = sin π₯.
Similarly, translating the sine function π
2 to the ________________________
results in the graph of the cosine function, π(π₯) = cos π₯.
Part 3: Even/Odd Function Identities Remember that ππ¨π¬ π is an __________ function. Reflecting its graph across the π¦-axis results in two equivalent functions with the same graph. sin π₯ and tan π₯ are both __________ functions. They have rotational symmetry about the origin.
Transformation Identities
Even/Odd Identities
Part 4: Co-function Identities The co-function identities describe trigonometric relationships between complementary angles in a right triangle.
We could identify other equivalent trigonometric expressions by comparing principle angles drawn in standard position in quadrants II, III, and IV with their related acute (reference) angle in quadrant I.
Principle in Quadrant II Principle in Quadrant III Principle in Quadrant IV
Example 2: Prove both co-function identities using transformation identities
a) cos (π
2β π₯) = sin π₯ b) sin (
π
2β π₯) = cos π₯
Co-Function Identities
Part 5: Apply the Identities
Example 3: Given that sinπ
5β 0.5878, use equivalent trigonometric expressions to evaluate the following:
b) cos7π
10
a) cos3π
10
1
x
y
L2 β 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(π₯ + π¦) β cos π₯ + cos π¦ Part 1: Proof of ππ¨π¬(π + π) and π¬π’π§(π + π) So what does cos(π₯ + π¦) =? Using the diagram below, label all angles and sides:
cos(π₯ + π¦) = sin(π₯ + π¦) =
Part 2: Proofs of other compound angle formulas Example 1: Prove cos(π₯ β π¦) = cos π₯ cos π¦ + sin π₯ sin π¦
Example 2: a) Prove sin(π₯ β π¦) = sin π₯ cos π¦ β cos π₯ sin π¦
LS
RS
LS = RS
Even/Odd Properties cos(βπ₯) = cos π₯ sin(βπ₯) = βsin π₯
LS
RS
LS = RS
1
1β2
2
1
β3
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(π₯ + π¦) = sin π₯ cos π¦ + cos π₯ sin π¦
sin(π₯ β π¦) = sin π₯ cos π¦ β cos π₯ sin π¦
cos(π₯ + π¦) = cos π₯ cos π¦ β sin π₯ sin π¦
cos(π₯ β π¦) = cos π₯ cos π¦ + sin π₯ sin π¦
tan(π₯ + π¦) =tan π₯ + tan π¦
1 β tan π₯ tan π¦
tan(π₯ β π¦) =tan π₯ β tan π¦
1 + tan π₯ tan π¦
a) sinπ
12
sinπ
12=
b) tan (β5π
12)
tan (β5π
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(π + π), where π and π are both angles in the second quadrant; given sin π =3
5 and
sin π =5
13
Start by drawing both terminal arms in the second quadrant and solving for the third side.
L3 β 4.5 Double Angle Formulas MHF4U Jensen
Part 1: Proofs of Double Angle Formulas Example 1: Prove sin(2π₯) = 2 sin π₯ cos π₯
Example 2: Prove cos(2π₯) = cos2 π₯ β sin2 π₯
Note: There are alternate versions of πππ 2π₯ where either πππ 2 π₯ ππ π ππ2 π₯ are changed using the Pythagorean Identity.
LS
RS
LS
RS
Part 2: Use Double Angle Formulas to Simplify Expressions Example 1: Simplify each of the following expressions and then evaluate
a) 2 sinπ
8cos
π
8
b) 2 tan
π
6
1βtan2π
6
Double Angle Formulas
sin(2π₯) = 2 sin π₯ cos π₯
cos(2π₯) = cos2 π₯ β sin2 π₯
cos(2π₯) = 2 cos2 π₯ β 1
cos(2π₯) = 1 β 2 sin2 π₯
tan(2π₯) =2 tan π₯
1 β tan2 π₯
Part 3: Determine the Value of Trig Ratios for a Double Angle If you know one of the primary trig ratios for any angle, then you can determine the other two. You can then determine the primary trig ratios for this angle doubled.
Example 2: If cos π = β2
3 and
π
2β€ π β€ π, determine the value of cos(2π) and sin(2π)
We can solve for cos(2π) without finding the sine ratio if we use the following version of the double angle formula:
Now to find sin(2π):
Example 3: If tan π = β3
4 and
3π
2β€ π β€ 2π, determine the value of cos(2π).
We are given that the terminal arm of the angle lies in quadrant ___:
L4β4.5ProveTrigIdentitiesMHF4UJensenUsingyoursheetofallidentitieslearnedthisunit,proveeachofthefollowing:Example1:Prove !"#(%&)
()*+!(%&)= tan π₯
Example2:Provecos 4
%+ π₯ = βsin π₯
LS
RS
LS
RS
Example3:Provecsc(2π₯) = *!* &% *+! &
Example4:Provecos π₯ = (
*+! &β sin π₯ tan π₯
LS
RS
LS
RS
Example5:Provetan(2π₯) β 2 tan 2π₯ sin% π₯ = sin 2π₯
Example6:Prove*+!(&9:)
*+!(&):)= ();<#& ;<#:
(9;<# & ;<#:
LS
RS
LS
RS
L5β5.4SolveLinearTrigonometricEquationsMHF4UJensenInthepreviouslessonwehavebeenworkingwithidentities.IdentitiesareequationsthataretrueforANYvalueofπ₯.Inthislesson,wewillbeworkingwithequationsthatarenotidentities.Wewillhavetosolveforthevalue(s)thatmaketheequationtrue.Rememberthat2solutionsarepossibleforananglebetween0and2πwithagivenratio.UsethereferenceangleandCASTruletodeterminetheangles.Whensolvingatrigonometricequation,considerall3toolsthatcanbeuseful:
1. SpecialTriangles2. GraphsofTrigFunctions3. Calculator
Example1:Findallsolutionsforcos π = β *+intheinterval0 β€ π₯ β€ 2π
Example2:Findallsolutionsfortan π = 5intheinterval0 β€ π₯ β€ 2πExample3:Findallsolutionsfor2 sin π₯ + 1 = 0intheinterval0 β€ π₯ β€ 2π
Example4:Solve3 tan π₯ + 1 = 2,where0 β€ π₯ β€ 2π
L6β5.4SolveDoubleAngleTrigonometricEquationsMHF4UJensenPart1:Investigation π = π¬π’π§π π = π¬π’π§(ππ)a)Whatistheperiodofbothofthefunctionsabove?Howmanycyclesbetween0and2πradians?b)Lookingatthegraphofπ¦ = sin π₯,howmanysolutionsarethereforsin π₯ = 2
3β 0.71?
c)Lookingatthegraphofπ¦ = sin(2π₯),howmanysolutionsarethereforsin(2π₯) = 2
3β 0.71?
d)Whentheperiodofafunctioniscutinhalf,whatdoesthatdotothenumberofsolutionsbetween0and2πradians?
Part2:SolveLinearTrigonometricEquationsthatInvolveDoubleAngles
Example1:sin(2π) = 93where0 β€ π β€ 2π
Example2:cos(2π) = β 23where0 β€ π β€ 2π
Example3:tan(2π) = 1where0 β€ π β€ 2π
L7β5.4SolveQuadraticTrigonometricEquationsMHF4UJensenAquadratictrigonometricequationmayhavemultiplesolutionsintheinterval0 β€ π₯ β€ 2π.Youcanoftenfactoraquadratictrigonometricequationandthensolvetheresultingtwolineartrigonometricequations.Incaseswheretheequationcannotbefactored,usethequadraticformulaandthensolvetheresultinglineartrigonometricequations.YoumayneedtouseaPythagoreanidentity,compoundangleformula,ordoubleangleformulatocreateaquadraticequationthatcontainsonlyasingletrigonometricfunctionwhoseargumentsallmatch.Rememberthatwhensolvingalineartrigonometricequation,considerall3toolsthatcanbeuseful:
1. SpecialTriangles2. GraphsofTrigFunctions3. Calculator
Part1:SolvingQuadraticTrigonometricEquations
Example1:Solveeachofthefollowingequationsfor0 β€ π₯ β€ 2π
a) sin π₯ + 1 sin π₯ β ,-= 0
b)sin- π₯ β sin π₯ = 2
c)2sin- π₯ β 3 sin π₯ + 1 = 0
Part2:UseIdentitiestoHelpSolveQuadraticTrigonometricEquationsExample2:Solveeachofthefollowingequationsfor0 β€ π₯ β€ 2πa)2sec- π₯ β 3 + tan π₯ = 0
b)3 sin π₯ + 3 cos(2π₯) = 2
L8β5.4ApplicationsofTrigonometricEquationsMHF4UJensenPart1:ApplicationQuestions
Example1:Today,thehightideinMatthewsCove,NewBrunswick,isatmidnight.Thewaterlevelathightideis7.5m.Thedepth,dmeters,ofthewaterinthecoveattimethoursismodelledbytheequation
π π‘ = 3.5 cos *+π‘ + 4
Jennyisplanningadaytriptothecovetomorrow,butthewaterneedstobeatleast2mdeepforhertomaneuverhersailboatsafely.DeterminethebesttimewhenitwillbesafeforhertosailintoMatthewsCove?
Example2:Acityβsdailytemperature,indegreesCelsius,canbemodelledbythefunction
π‘ π = β28 cos 1*2+3
π + 10
whereπisthedayoftheyearand1=January1.Ondayswherethetemperatureisapproximately32Β°Corabove,theairconditionersatcityhallareturnedon.Duringwhatdaysoftheyeararetheairconditionersrunningatcityhall?
Example3:AFerriswheelwitha20meterdiameterturnsonceeveryminute.Ridersmustclimbup1metertogetontheride.a)Writeacosineequationtomodeltheheightoftherider,βmeters,π‘secondsaftertheridehasbegun.Assumetheystartattheminheight.b)Whatwillbethefirst2timesthattheriderisataheightof5meters?
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