Section6:TrigonometricIdentitiesandApplicationsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).6.01TrigonometricIdentities
• Precalculus(5)(M)• Precalculus(5)(N)
6.02SolvingTrigonometricEquations
• Precalculus(5)(M)• Precalculus(5)(N)
6.03SumandDifferenceFormulasofTrigonometricFunctions
• Precalculus(5)(M)• Precalculus(5)(N)
6.04DoubleAngleandHalfAngleFormulasofTrigonometricFunctions6.05LawofSinesandCosines
• Precalculus(4)(G)• Precalculus(4)(H)
6.06TrigonometricWordProblems
• Precalculus(4)(E)• Precalculus(4)(F)• Precalculus(5)(M)• Precalculus(5)(N)
6.07Vectors
• Precalculus(4)(I)• Precalculus(4)(J)• Precalculus(4)(K)
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6.01TrigonometricIdentities
ReciprocalIdentities
sin𝜃 = '()( *
cos𝜃 = ')-(*
tan𝜃 = '(01 *
QuotientIdentities
tan𝜃 = )23*(0)*
cot𝜃 = (0)*)23 *
CofunctionIdentities
sin𝜃 = cos 90 − 𝜃 sec𝜃 = csc 90 − 𝜃 tan𝜃 = cot 90 − 𝜃
cos𝜃 = sin 90 − 𝜃 csc𝜃 = sec 90 − 𝜃 cot𝜃 = tan 90 − 𝜃
NegativeAngleIdentities(Even/OddFunctions)
sin −𝜃 = −sin𝜃
cos −𝜃 = cos𝜃
tan −𝜃 = −tan𝜃
PythagoreanIdentities
sin8𝜃 + cos8𝜃 = 1
1 + tan8𝜃 = sec8𝜃
1 + cot8𝜃 = csc8𝜃
1. Givensec ;8− 𝜃 = 3andcos 𝜃 > 0,findcot 𝜃.
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Belowaresomeguidelinesformatchingorverificationproblems:
Step1: Startwithanyidentities.
Step2:Usethemethodsoffactoring,commondenominators,separatingnumerators,andconjugates.
Step3: Ifnothingisworking,trychangingeverythingtosin𝜃andcos𝜃.
2. Verify'>(0) ?'@(0) ?
= 2csc8𝑥 − 2 csc 𝑥 cot 𝑥 − 1.
3. Verify()( ?@'()( ?>'
= '@)23 ?'>)23 ?
.
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6.02SolvingTrigonometricEquations
Fortheseequations,morethanonesolutionmayexist,ortheremaybenosolution.
1. Solveforallangles𝐴:2 cos 𝐴 − 1 = 0
Belowarestepstousewhenaskedtosolveforallangles:
Step1: Solveforangleswithinthespecifiedinterval.
Step2: Iftheanglesareseparatedby𝜋radians,takethesmallestofthetwoanglesandadd𝜋𝑛,where𝑛isaninteger.
Step3: Iftheanglesarenotseparatedby𝜋radians,takeeachangleandadd2𝜋𝑛,where𝑛isaninteger.
2. Solvecos 𝑥 cot 𝑥 = cos 𝑥forallangles.
3. Solvecos8 𝐴 − 4 cos𝐴 + 3 = 0intheinterval[0,2𝜋).
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6.03SumandDifferenceIdentities
sin 𝑢 + 𝑣 = sin𝑢cos𝑣 + cos𝑢sin𝑣 sin 𝑢 − 𝑣 = sin𝑢cos𝑣 − cos𝑢sin𝑣
cos 𝑢 + 𝑣 = cos𝑢cos𝑣 − sin𝑢sin𝑣 cos 𝑢 − 𝑣 = cos𝑢cos𝑣 + sin𝑢sin𝑣
tan 𝑢 + 𝑣 = 1L3M@1L3N'>1L3M1L3N
tan 𝑢 − 𝑣 = 1L3M>1L3N'@1L3M1L3N
1. Findcos 75°usingsumanddifferenceidentities.
2. Findtan 𝐴 + 𝐵 ifsin𝐴 = ST,𝐴isinQuadrantII,sin𝐵 = − '8
'U,and𝐵isinQuadrantIV.
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6.04Double-AngleandHalf-AngleIdentities
Double-AngleFormulas
sin 2𝑢 = 2sin𝑢cos𝑢cos 2𝑢 = cos8𝑢 − sin8𝑢 = 2cos8𝑢 − 1 = 1 − 2sin8𝑢
tan 2𝑢 =2tan𝑢
1 − tan8𝑢
1. Ifsin𝐵 = − 'Uandtan𝐵 < 0,findsin 2𝐵.
2. Solvesin 2𝐴 = cos𝐴ontheinterval[0,2𝜋).
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Half-AngleFormulas
sin
𝑢2 = ±
1 − cos 𝑢2
cos𝑢2 = ±
1 + cos 𝑢2
tan𝑢2 = ±
1 − cos 𝑢1 + 𝑐𝑜𝑠𝑢 =
1 − cos 𝑢sin 𝑢 =
sin 𝑢1 + cos 𝑢
3. Findtheexactvalueofsin 105°usinghalf-angleformulas.
4. Findtheexactvalueofcos ?8ifcos 𝑥 = '
Tand𝑥isinthefourthquadrant.
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6.05LawofSinesandCosines
Thelawofsinesandcosinesisusedforobliquetriangles,whicharetrianglesthatdonothaverightangles.
Givenangles𝐴,𝐵,and𝐶ofanobliquetrianglewithoppositesides𝑎,𝑏,and𝑐,wehave
Lawofsines–)23^_= )23`
a= )23 b
c
Lawofcosines–𝑎8 = 𝑏8 + 𝑐8 −2𝑏𝑐 cos 𝐴 𝑏8 = 𝑎8 + 𝑐8 −2𝑎𝑐 cos𝐵
𝑐8 = 𝑎8 + 𝑏8 −2𝑎𝑏 cos 𝐶
Herearesomestrategiestoconsiderwhendecidingtousethelawofsinesorthelawofcosines:
• Lawofsines:Usewhenyouaregiventwoanglesandoneside.• Lawofcosines:Usewhenyouaregivenallthreesidesortwo
sidesandoneangle.
1. Givenanobliquetrianglewhere𝑎 = 4,𝑏 = 7,and𝑐 = 6,findangle𝐶.
2. Givenanobliquetrianglewhere𝑏 = 4,𝑐 = 7,andangle𝐶 = 50,findsin𝐵.
3. Findside𝑐ofanobliquetrianglewithside𝑎 = 4,angle𝐴 = 50°,andangle𝐵 = 60°.
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Wehavetwoequationsfortheareaofanobliquetriangle:
Heron’sformulaforarea–Givensides𝑎,𝑏,𝑐andthesemi-perimeter𝑠,wehave
( )12
s a b c= + + andthearea ( )( )( )A s s a s b s c= - - -
Areaofanobliquetriangle–Giventwosidesandtheanglebetweenthem,wehave
𝐴 = '8𝑎𝑏 sin 𝛾 𝐴 = '
8𝑎𝑐 sin 𝛽 𝐴 = '
8𝑏𝑐 sin 𝛼
4. Findtheareaofanobliquetrianglewiththeproperties𝑎 = 4,𝑏 = 7,and𝐶 = 75°.
5. Findtheareaofatrianglewhosesideshavelengthsof3ft,5ft,and6ft.
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6.06TrigonometricWordProblems
Herearetwotipswhensettingupwordproblemsinvolvingtrigonometry:
• Angleofelevationandangleofdepressionarewithrespecttothe𝑥-axis.
• Drawapicture!
1. Supposeyouleanan8-footladderagainstawallata60-degreeangleofelevation.Howhighisthetopoftheladderfromthebaseofthewall?
2. Imaginethatamanisfloatingintheocean,waitingtoberescued.Hespotstwohelicoptersonexactlyoppositesidesofhim,oneata30-degreeangleofelevationandtheotherata60-degreeangleofelevation.Ifthehelicoptersarebothflyingatanaltitudeof100meters,whatisthedistancebetweenthehelicopters?
3. Aplaneliftsoffatanangleofelevationof20degreesatarateof300feetpersecond.Howmanyminutespassbeforeitreachesanaltitudeof6,000feet?
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6.07Vectors
Avectorisadirectedlinesegmentinaplane.
• Avectorisadirectedlinesegmentandhasaninitialpointandaterminalpoint.
• Vectorshaveamagnitude(length)andadirection.
o Magnitudecanbecalculatedusingthedistanceformula.
o Directioncanbecalculatedusingtrigonometry.
• Vectorscanbewrittenas𝑣 =𝑃𝑄where𝑃and𝑄aretwopointsonaplane.
• Vectorscanalsobewrittenincomponentformas𝑣 =𝑣', 𝑣8 .
1. Findthecomponentformandmagnitudeofavectorwithinitialpoint(2,−5)andterminalpoint(−1,−9).
2. Ashipsailsfromportandtravelsonabearingof30degreesnorthofeastataspeedof20nauticalmilesperhour.Afterthreehours,howfareasthastheshipsailed?
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Vectoraddition–Giventwovectors𝑢 = 𝑢', 𝑢8 and𝑣 = 𝑣', 𝑣8 ,then𝑢 + 𝑣 = 𝑢' + 𝑣', 𝑢8 + 𝑣8 .
Scalarmultiplication–Givenavector𝑣 = 𝑣', 𝑣8 andaconstant𝑐,then𝑐𝑣 = 𝑐𝑣', 𝑐𝑣8 .
3. Given𝑢 = 5,8 and𝑣 = −3,7 ,find𝑢 + 𝑣,3𝑢,and3𝑢 − 4𝑣incomponentform.
4. Acycliststartedfromherhomeandtraveledfourmileswest,thenthreemilesnortheast,andfinallytwomilessouth.Howfaristhecyclistfromherhome?
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