c

a

b

A

B

C

MCR3U–Unit5:TrigonometricRatios–Lesson6 Date:___________Learninggoal:IcansolveforanunknownangleorsideofanobliquetriangleusingtheSineandCosineLaw.

SineandCosineLawRECALL:upper-caselettersidentifiestheverticesofatriangleandequivalentlower-caselettersidentifiesthesidesoppositevertices.

Thesidelengthsoftrianglesarerelatedtotheiranglemeasurementsthroughtrigonometricratios.

Example1:Solvethefollowingtriangle.

Example2:Solvefor∠!.

Therearesomeproblemswiththesinelaw.

Thesewillbediscussedlaterinourunitduringourlessonontheambiguouscase.

68.5 m4263

12.3 cm

9.1 cmA

120

TheSineLaw

or

COSINELAW

Ifyouaregiventwosidesandthecontainedangle,orthelengthofallthreesides,youcanusethecosinelaw.TheCosineLawonlyneedstobeusedonceinsideatriangle.AftertheuseoftheCosineLaw,youareguaranteedtohaveanangle-sidepair,andcanthereforeusetheSineLaw.

Example3:Solvefor∠!.

Example4:Threefriendsarecampinginthewoods,Bert,ErnieandElmo.Theyeachhavetheirowntentand

thetentsaresetupinaTriangle.BertandErnieare10mapart.TheangleformedatBertis30°.Theangle

formedatElmois105°.HowfarapartareErnieandElmo?

HW:Pg.509#[1-4,7]partconly,11,15,17,23

TheCosineLaw RearrangedtoSolvefortheAngle

*Thesidedesignated mustbeoppositetheangle

3.0 mm

2.0 mm

4.0 mm

A

MCR3U–Unit5:TrigonometricRatios–Lesson7 Date:___________Learninggoal:Icanidentifytheambiguouscaseanddeterminethenumberofsolutionsinatriangle.

TheAmbiguousCaseRECALL:Supplementaryanglesaddto180°.WARMUP:Calculatethefollowing,roundingto4decimalplaces.

!"#50° =_________________ !"#!! (0.6428) =_________________!"#130° =________________ !"#!! (−0.6428) =________________

!"#50° =_________________ !"#!! (0.7660) =_________________!"#130° =________________ !"#!! (0.7660) =________________

CONCLUSION

Thesineratiosforanacuteangleanditsobtusesupplementarethe____________________andthe

_______________________.However,whentakingtheinversesineoftheratiotheanglereturnedis

alwaysthe_____________________________________.Thismeansthesineratiois_________________.

Thecosineratiosforanacuteangleanditsobtusesupplementarethe____________________but

________________________________.Thus,thecorrectangleisreturnedwhenyoutaketheinverse

cosineoftheratio.Thismeansthecosineratioisnot____________________.

RECOGNIZINGTHEAMBIGUOUSCASE

Inallcasesbelowconsideryouhaveananglesidepair(∠!, !"#$ !)andanotherside(!"#$ !).

1. Case1:∠! ≥ !"°,! > !ThereisONEsolution

2. Case2:∠! ≥ !"°,! ≤ !ThereisNOsolution

3. Case3:∠! < !"°,! > !ThereisONEsolution

4. Case4: ∠! < !"°,! ≤ !ThereareTHREEpossiblesolutions

a) If! = !"#$!% !" !"# !"#$%&'(Tofindtheheight….!"#$ = !

!

!(!"#$) = !∴if! = !(!"#$)thereisONEsolution

b) If! < !"#$!% !" !"# !"#$%&'(Tofindtheheight….!"#$ = !

!

!(!"#$) = !∴if! < !(!"#$)thereisNOsolution

c) If! > !"#$!% !" !"# !"#$%&'(Tofindtheheight….!"#$ = !

!

!(!"#$) = !∴if! > !(!"#$)thereisTWOsolutions

SUMMARY

Thesinelawmaybeambiguouswhenyouhaveaside-anglepairandlargersideorequalside.

ItisNOTambiguouswhenyouhaveasideanglepairandasmallerside.

∠!

∠! < !"°

! > ! ! ≤ !

! > !"#$!%! < !"#$!%! = !"#$!%

!"# !"#$%&"' !" !"#$%&"' !"# !"#$%&"'!

∠! ≥ !"°

! > ! ! ≤ !

!" !"#$%&"'!"# !"#$%&"'

Example1:Determinethenumberoftrianglesthatcouldbedrawgiven∆!"#,where∠! = 45°,! = 30!", ! = 24!".

Example2:Determinethenumberoftrianglesthatcouldbedrawgiven∆!"#,where∠! = 40.3°,! = 35.2!", ! = 40.5!".

Example3:Determinethenumberoftrianglesthatcouldbedrawgiven∆!"#,where∠! = 143°,! = 12.5!", ! = 8.9!".

HW:TheAmbiguousCaseWorksheet

Ambiguous Case Number of Solutions WorksheetMCR3U 4.7.1

M

O

N

B

C

A

20

127

7

95 °

10

P

Q

R

19

20

100 °

40 °

RT

S

L

M

N

20

11

B

A

C

18

15

30 °

40 °

T

R

S

150 °

17

J

LK

8

40 °

10

J

LK

12

30 °

E

G

F

120 °50

40

30

30 °

R

TS

910

60 °

A

C

B

8.5

10

110

115

1. Determine the number of solutions for each, with adequate evidence and then

actually solve the triangles that actually have solutions.

1

2

3

4

5

6

7

8

9

10

11

1250 °

MCR3U–Unit5:TrigonometricRatios–Lesson8 Date:___________Learninggoal:Icanidentifyandsolveanambiguoustriangle.

SolvingAmbiguousCasesYesterdaywelearnedaboutambiguoustriangles.Todaywearegoingtosolveambiguoustriangles.

Example1:Solve∆!"#,where∠! = 44.3°,! = 7.7!",! = 11.5!".

Example2:Solve∆!"#,where∠! = 29.3°,! = 20.5!", ! = 12.8!".

Example2:Solve∆!"#,where∠! = 38.7°,! = 10!", ! = 25!".

HW:Pg.511#5,6(i,iii,iv),8(AnsCorr5c:2solutions)

MCR3U–Unit5:TrigonometricRatios–Lesson9 Date:___________Learninggoal:Icanunderstanddirectionsstatedinawordproblem.Icandrawandsolve2Dapplicationproblems.

2DApplications

Angleofelevation/inclination:Theangleofalineofsight,measuredupfromthehorizontal.

Angleofdepression:Theangleofalineofsight,measureddownfromthehorizontal.

CompassDirections:

North,South,East,West

Northeast,Northwest,Southeast,Southwest

Relativecompassdirections:Anangleisusedtodescribedirection,startingatonecompassdirectionandrotatingtowardanothercompassdirection

Bearing:Anangleusedtodescribeadirection,measuredrelativetocompassnorth.Theyaremeasuredinaclockwisedirection.

Heading:Anangleusedtodescribeadirection,measuredrelativetoalinedirectlyinfrontofanobject.

SuccessCriteriafor2DApplications

1. Createasketch

2. Underline/highlight/writedownimportantinformation

3. Checktheambiguouscase

4. Addnewinformationtoyoursketchasyousolveforunknowns

5. Concludingsentence

Example1:Aprisontowerhasaspotlightataheightof60m.Itprojectsaconeoflighttothegroundbelow.Theangleofdepressiontothetopoftheconeis18!.Theangleofdepressiontothebottomoftheconeis21.8!.Findthewidthofthespotoflightontheground(atitslargest).

Example2:Adestroyerandacarrierarehuntinganenemysubmarine.Theyare10kmapart,withthecarrieratabearingof20!fromthedestroyer.Thedestroyerdetectsthesubatabearingof70!,butcannotestablishadistance.Thecarrierdetectsthesubatadistanceof8km,butcannotestablishadirection.Howfaristhesubfromthedestroyer?

Example3:AnairplaneleavesCityAandfliesS25oWat800km/hforonehourtoreachCityB.ItthenheadsS30oEandfliesatthesamespeedfortwoandahalfhourstoreachCityC.HowfarisCityCfromCityA?

HW: Pg.522#1,4-6,9,14-15

MCR3U–Unit5:TrigonometricRatios–Lesson10 Date:___________Learninggoal:Icandrawthensolve3Dapplicationproblems.

3DApplications

Thetricktosolving3Dwordproblemsistobreakthemdownintomultiple2Dwordproblems.Agoodsketchisalsouseful.

Example1:FrompointB,Mannyusesaclinometertodeterminetheangleofelevationtothetopofacliffis38!.FrompointD,68.5mawayfromManny,Joeestimatestheanglebetweenthefootofthecliff,himself,andMannytobe42!,whileMannyestimatestheanglebetweenthefootofthecliff,himself,andJoetobe63!.Whatistheheightoftheclifftothenearesttenthofametre?

Example2:Carlisona50mhighbridgeandseestwoboatsanchoredbelow(theArgoandtheBluenose).Fromhisposition,theArgohasabearingof230!andtheBluenosehasabearingof120!.Carlestimatestheanglesofdepressiontobe38!totheArgoand35!totheBluenose.Howfarapartaretheboatstothenearestmetre?

HW:3DApplicationsWorksheet

3DApplicationsWorksheet

1. Philipisflyingahot-airballoonfromBeamsvilletoVineland.Hedecidestocalculatethe

straight-linedistance,tothenearestmetre,betweenthetwotowns.Fromanaltitudeof226metreshemeasurestheangleofdepressiontoBeamsvilleas2!andtoVinelandas3!.Healsomeasurestheanglebetweenthelinesofsighttothetwotownsas80!.HelpPhilipcalculatethedistancefromBeamsvilletoVineland.

2. RomeoisstandingdirectlysouthofJuliet’sbalcony.Hemeasurestheangleofelevationtothebalconytobe20!.ParisisstandingdirectlyeastofJuliet’sbalcony.Hemeasurestheangleofelevationtothebalconytobe18!.IfRomeoandParisare100mapart,findtheheightofJuliet’sbalcony,tothenearesttenthofametre.

3. Simoneisfacingnorthattheentranceofatunnelthroughamountain.Shenoticesthata1515mhighmountaininthedistancehasabearingof270!anditspeakappearsatanangleofelevationof35!.Aftersheexitsthetunnel,thesamemountainhasabearingof258!.Assumingthatthetunnelisperfectlylevelandstraight,howlongisittothenearestmetre?

4. Anairportradaroperatorlocatestwoplanesflyingtowardtheairport.Thefirstplaneisaboveapoint12kmfromtheairportatabearingof70!andwithanaltitudeof10.7km.Theotherplaneisaboveapoint18kmawayonabearingof125!andwithanaltitudeof1.5km.Calculatethedistancebetweenthetwoplanestothenearesttenthofakilometre.(Hint:Thefirsttwodistancesarehorizontal.Thedistanceyouarefindingisnot).

Answers

1.ThereforeBeamsvilleandVinelandareabout7127mapart.

2.Thereforethebalconyisabout24.2mtall.

3.Thereforethetunnelisabout460mlong.

4.Thereforetheplanesareabout17.4kmapart.