Download 9D: Multiple-Angle Identities

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9D:Multiple-AngleIdentities
Theangle-sumanddifferenceidentitiesopenthedoortoseveralgroupsofusefulidentities.Inthis
lesson,wewillseehowwecanusetheanglesumidentitiestoderivethesenewidentities.
Remember:
sin!" + $% = sin " cos $ + cos " sin $ ,
cos!" + $% = cos " cos $ − sin " sin $,
tan!" + $% =
DoubleDouble-AngleIdentities
AngleIdentities
tan " + tan $
1 − tan " tan $
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Since!" + "% = 2",wecanusetheangle-sumidentitiestofindasimplifiedexpressionforfunctions
oftheangle2".Let’stryit!Simplifyeachofthefollowingexpressionsusingangle-sumidentitiesso
thatweareleftwithfunctionsoftheangleu.
a% sin 2" =
b% cos 2" =
c% tan 2" =
Inpart!b%youshouldhavefoundthatcos 2" = cos 1 " − sin1 ".However,wealsoknowthe
Pythagoreanidentitycos 1 " + sin1 " = 1.
d% UsethePythagoreanidentitytowritecos 2"intermsofsin ".
e% UsethePythagoreanidentitytowritecos 2"intermsofcos ".
DoubleDouble-AngleIdentities
AngleIdentities
sin 2" = 2 sin " cos "
cos 2" = cos " − sin "
cos 2" = 2 cos1 " − 1
cos 2" = 1 − 2 sin1 "
1
1
tan 2" =
2 tan "
1 − tan1 "
ExampleProvetheidentitycos 6 7 − sin6 7 = cos 27
PowerPower-ReducingIdentities
ReducingIdentities
Sometimeswehaveapowerofatrigonometricfunctionlikesin1 "thatweneedtowritewithoutan
exponent.Todothis,weneedsomethingthatwecall“power-reducingidentities”.
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Usethesecondtwocosinedouble-angleidentitiestofindexpressionsforsin1 ",cos1 ",andtan1 ".
a% Solveforsin1 ":
cos 2" = 1 − 2 sin1 "
b% Solveforcos1 ":
cos 2" = 2 cos1 " − 1
:;<= >
c% Use!a%and!b%aboveandthefactthattan1 " = ?@:= >tofindanexpressionfortan1 "in
termsofcos 2":
tan1 " =
PowerPower-ReducingIdentities
ReducingIdentities
1 − cos 2"
sin1 " =
2
Example:
1 + cos 2"
cos1 " =
2
tan1 " =
1 − cos 2"
1 + cos 2"
Rewritesin6 7intermsoftrigonometricfunctionswithnopowergreaterthan1.
HalfHalf-AngleIdentities
AngleIdentities
Thepowerreducingidentitiesgiveusanequationthatstartswithanangle"andproducesanew
C
expressionintermsof2".Whathappensifwelet= 1 ?Thesinepower-reducingidentitygivesus
F
1 − cos H2 ⋅ 2 J
F
sin1 E G =
2
2
F
1 − cos F
sin E G = ±L
2
2
C
a% Usethesameprocesstofindanidentityforcos H JintermsofFusingthepower-reducing
1
formulas.
C
b% Usethesameprocesstofindanidentityfortan H1 JintermsofFusingthepower-reducing
formulas.
HalfHalf-Angle
AngleIdentities
Identities
Identities
"
1 − cos "
sin H J = ±L
2
2
"
1 + cos "
cos H J = ±L
2
2
"
1 − cos "
tan H J = ±L
2
1 + cos "
"
1 − cos "
tan H J =
2
sin "
"
sin "
tan H J =
2
1 + cos "
SolvingEquations
SolvingEquations
Whensolvingtrigonometricequationswiththeseidentities,thegeneralstrategyisto
1%
2%
3%
4%
Rewritetheexpressionsintermsofasingleangle!notahalformultipleangle%,
Settheequationequalto0,
Factorthenon-zeroside,and
Usethezeroproductruletosolve.
TryIt:Findallsolutionstotheequationontheinterval[0,2S%.
a% sin 2F = sin F
T
b% sin1 7 = cos1 H1J
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Assignment9D:Multiple-AngleIdentities
1. Usetheappropriatesumordifferenceidentitytoprovethedouble-angleidentity.
cos 27 = cos 1 7 − sin1 7
2. Usetheresultfrom#1toshowthefollowing:cos 27 = 2 cos 1 7 − 1
3. Usetheresultfor#2toderiveaformulaforcos 1 7intermsofcos 27
>
4. Finally,usetheresultfrom#3toderiveaformulaforcos H Jintermsofcos ".
1
>
!Hint:beginbyletting7 = 1 .%
Findallthesolutionstotheequationintheinterval[0,2S%
5. sin 27 = cos 7
6. cos 27 = sin 7
7. cos1 7 + cos 7 = cos 27
8. Writetheexpressionasoneinvolvingonlysin Fandcos F.sin 2F + cos 3F
Provetheidentity.
Provetheidentity.
9. sin 47 = 2 sin 27 cos 27
10. 2 csc 27 = csc 1 7 tan 7
11. sin 37 = !sin 7%!4 cos1 7 − 1%
Usehalf-angleidentitiestofindanexactvaluewithoutacalculator.
12. sin 15°
13. cos 75°
[
14. tan \