6.2 Trig.FunctionsofAcuteAnglesPre-Calculus

Next let’s put our anglesinsidetriangles…specificallyright triangles.

Three Basic Trigonometric Ratios

sinsideopposite,cossideadjacentto,tansideopposite

hypotenusehypotenusesideadjacentto

  • Remembertheratiosareusedonanacuteangle.
  • Wememorized.

The Reciprocal Ratios:,and

csc

1

sin

sec

1

cos

cot

1

tan

Example1:Usingthetriangleattheright,findallsixtrigonometricfunctionsoftheangle.

21

Example2:Giventan= 5

12

,findtheremainingtrigonometricfunctions.

SpecialRightTriangles:

#1:45 – 45 – 90 Right Triangle(MEMORIZETHESERATIOS)A

a)IfAC=1,andmA=45,solvefortheremainingpartsofthetriangle.

BC

b)Findthesine,cosineandtangentvaluesof45.

A

#2:30-60–90Right Triangle(MEMORIZETHESERATIOS)

a)DABC

isequilateral,soeachangleis.

b)DrawthealtitudeofthetrianglefromAtoBC.CallthepointofintersectionD.

c)Therefore,mBAD=.

d)SupposeAB=2.Solveforthe remainingpartsofDBAD.

BC

e)Findthesine,cosineandtangentvaluesof30and60.

Example4:Aladderisextendedtoreachthetopfloorofan84foottallburningbuilding.Thefirefightersseesomeonewhoneedsrescuinginawindow8feetbelowtheroof.Howfarshouldtheladderbeextendedtoreachtheroofiftheladdermustbeplacedattheoptimumoperatingangleof60?

6.4 InverseTrigonometric FunctionsPre-Calculus

InverseSineInverseCosineInverseTangent

ysin1xyarcsinx

ycos1xyarccosx

ytan1xyarctanx

D:[–1,1]D:[–1,1]D:(–∞,∞)

R: R: R:



Example1:Findtheexactvalue(inradians).

a) cos-10b)sin-10c)d)

e) f) g) h)

Workfromtheinsideout.

Rememberdomainandrangerestrictions.

Example2:Evaluateeachexpression.

a) b)

Example3:Findthealgebraicexpressionequivalenttothegivenexpression.

a)sin(cos1 x)b) cot(sin12x)

Refer to pages 464 -465 Examples 4 and 5

SolvingProblemswithTrigonometryPre-Calculus

Nextwewillapplywhatweknowaboutthetrigonometricfunctions,andtheirinverses,tosolverealworldapplicationproblems.Themostimportantpartofthesetypesofquestionsisanaccurate,detailedpicture.

AngleofElevation:TheacuteanglemeasuredfromahorizontallineUPtoanobject.

AngleofDepression

AngleofDepression:TheacuteanglemeasuredfromahorizontallineDOWNtoanobject.

AngleofElevation

Example1:Fromaboatonthelake,theangleofelevationtothetopofacliffis2042.Ifthebaseofthecliffis1394feetfromtheboat,howhighisthecliff(tothenearestfoot)?