1 Trigonometry and Right Angles - Solution Key

257 Pages·2012·5.2 MB·English

Trigonometry Teacher’s
Edition - Solution Key
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1 TrigonometryandRightAngles-SolutionKey 1
1.1 TrigonometryandRightAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 CircularFunctions-SolutionKey 28
2.1 CircularFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 TrigonometricIdentities-SolutionKey 44
3.1 TrigonometricIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 InverseFunctionsandTrigonometricEquations-SolutionKey 96
4.1 InverseFunctionsandTrigonometricEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 TrianglesandVectors-SolutionKey 129
5.1 TrianglesandVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 PolarEquationsandComplexNumbers-SolutionKey 224
6.1 PolarEquationsandComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
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C 1
HAPTER
Trigonometry and Right
Angles - Solution Key
CHAPTER OUTLINE
1.1 TRIGONOMETRY AND RIGHT ANGLES
CHAPTER1. TRIGONOMETRYANDRIGHTANGLES-SOLUTIONKEY
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1.1
Trigonometry and Right Angles
Basic Functions
ReviewExercises:
1. a)Thisrelationisnotafunction. Thex−valueof1ispairedwithtwoy−values: 5and7.
b)Thisrelationisafunction. Anyverticallinewillcrossthegraphofy=3−xonlyonce. Eachx−valueispaired
withoneandonlyoney−value.
x −3 −2 −1 0 1 2 3
y 6 5 4 3 2 1 0
c)Thisrelationisnotafunction. Anyverticallinewillcrossthegraphmorethanonce.
2. a)
distance=rate·time
d=95t
b)Thissituationisdirectvariationbecauseasthetimeincreasesthedistanceincreasesatthesamerate.
c)
d=95t →generalequation
d=95miles/hr(3hr)→givent =3hours
d=285miles.
3. a)y=mx+b.yrepresentsthecostofbeginningthebusiness;mrepresentsthecostofeachwoodenframe(x)and
brepresentstheinitialoutputofmoney(0,100).
1.1. TRIGONOMETRYANDRIGHTANGLES
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y=2x+100
c(x)=2x+100→y=2x+100writtenasafunction.
b) y=mx+b. y represents the revenue; m represents the selling price of each picture frame and b represents any
otherrevenuewhichinthiscaseiszero.
y=10x
R(x)=10x→y=10xwrittenasafunction.
c)
P(x)=R(x)−C(x)→TheprofitP(x)isthedifferencebetweentherevenueR(x)andthecostC(x)
P(x)=10x−(2x+100)
P(x)=10x−2x−100
P(x)=8x−100
4. a)Thefunctiondefinedbytheequation f(x)=x2−x−3isofthegeneralformofaquadraticfunction.
b)Thedomainofthefunctionis: {xIxεR}
Therangeofthefunctionis: {yIy≥−3.25,yεR}
c)UsingtheTI–83tograph f(x)=x2−x−3
Thecoordinatesofthevertexandthex−interceptscanbedeterminedbyusingthe2nd Tracefunction:
Vertex
x−intercepts
CHAPTER1. TRIGONOMETRYANDRIGHTANGLES-SOLUTIONKEY
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Thevertexis(5.0,−3.25)andthex−interceptsare(−1.3,0)and(2.3,0).
5. a)UsingtheTI–83tography= x−2 :
x+3
Theasymptotesarex=−3andy=1
6. a)c= 1(500)→thecostperperson(c)ofrentingapartyroomvariesinverselywiththenumberofpeoplewho
p
attendandtheinitialcostofrentingtheroom(500)
b)
1
c= (500)→ given p=32
p
1
c= (500)
32
c=$15.63
7. a)UsingtheTI–83tograph:
y=x3
y=x3+x
y=x3+2x
Theequationswithpositivecoefficientslookmoreandmorelikey=x3 ,asthecoefficientgetslarger.
y=x3
y=x3−x
y=x3−2x
1.1. TRIGONOMETRYANDRIGHTANGLES
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Theequationswithnegativecoefficientshavelocalmaximumsandminimums.
Decreasingthecoefficientincreasesthesizeofthe”hill”andthe“valley.”
8. a)UsingtheTI–83tographthefunction p(x)=−.5x2+90x−200:
Thenumberofunitsthatmustbesoldtoattainthemaximumprofitisthevertexoftheparabola. Use2nd Trace
Themaximumprofitis$3850with90unitsbeingsold.
Thex−interceptsare:
and
CHAPTER1. TRIGONOMETRYANDRIGHTANGLES-SOLUTIONKEY
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Thex−interceptsrepresentthebreak-evenpointsofthecompany. Thecompanymustsellatleast2.25unitstocover
anyinitialcostsbutwhen177.7unitsaresold,itnolongermakesaprofit.
9. a)UsingtheTI–83tocreateascatterplotofthegivendata:
b)Theperiodistwelvemonths.
c) The number of daylight hours in other areas would not show as much variance, so the amplitude of the graph
wouldbesmaller.
Angles in Triangles
ReviewExercises:
1.
(cid:52)ABC isisosceles. Anisoscelestrianglehastwosidesequalinlength. ThereforeAC iseither5inchesinlengthor
7inchesinlength.
2. An obtuse triangle is one that has one angle that measures greater than 90◦ . A right triangle is one that has one
angle that measures 90◦ . The sum of the angles of a triangle is 180◦ . Therefore a right triangle has one angle of
90◦ andtwoacuteangles. Arighttrianglecannotbeanobtusetriangle.
3. Inanytriangle,thesumofthethreeanglesis180◦ .
(cid:54) 1+(cid:54) 2+(cid:54) 3=180◦
48◦+28◦+(cid:54) =180◦
76◦+(cid:54) 3=180◦
(cid:54) 3=180◦−76◦
(cid:54) 3=104◦
1.1. TRIGONOMETRYANDRIGHTANGLES