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Name: Date: Period: 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 $ Explore 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”. Explore 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 Name: Date: Period: 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 \