Section 6: Trigonometric Identities and Applications · 2018-02-01 · Section 6: Trigonometric...

Section6:TrigonometricIdentitiesandApplicationsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).6.01TrigonometricIdentities

• Precalculus(5)(M)• Precalculus(5)(N)

6.02SolvingTrigonometricEquations


6.03SumandDifferenceFormulasofTrigonometricFunctions


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)

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

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.


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 ?

.


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).


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.


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).


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.


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°.


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.


6.06TrigonometricWordProblems

Herearetwotipswhensettingupwordproblemsinvolvingtrigonometry:

• Angleofelevationandangleofdepressionarewithrespecttothe𝑥-axis.

• Drawapicture!

1. Supposeyouleanan8-footladderagainstawallata60-degreeangleofelevation.Howhighisthetopoftheladderfromthebaseofthewall?

2. Imaginethatamanisfloatingintheocean,waitingtoberescued.Hespotstwohelicoptersonexactlyoppositesidesofhim,oneata30-degreeangleofelevationandtheotherata60-degreeangleofelevation.Ifthehelicoptersarebothflyingatanaltitudeof100meters,whatisthedistancebetweenthehelicopters?

3. Aplaneliftsoffatanangleofelevationof20degreesatarateof300feetpersecond.Howmanyminutespassbeforeitreachesanaltitudeof6,000feet?


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?


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?


Section 6: Trigonometric Identities and Applications · 2018-02-01 · Section 6: Trigonometric...

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