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Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. g. 5cos2x+3cosx−2=0

Pertanyaan

Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0 degree less or equal than x less or equal than 360 degree.

g. 5 space cos squared space x plus 3 space cos space x minus 2 equals 0 

Pembahasan Soal:

Mencari himpunan penyelesaian:

5 space cos squared space x plus 3 space cos space x minus 2 equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell Misal space cos open parentheses x close parentheses end cell equals u row cell 5 u squared plus 3 u minus 2 end cell equals 0 row u equals cell 2 over 5 comma thin space u equals negative 1 end cell row cell cos open parentheses x close parentheses end cell equals cell 2 over 5 comma thin space cos open parentheses x close parentheses equals negative 1 end cell end table 
 

table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell x equals arccos open parentheses 2 over 5 close parentheses plus 360 degree n comma thin space end cell row x equals cell 66 comma 42182 degree plus 360 degree n end cell row blank blank blank row blank rightwards arrow cell x equals 360 degree minus arccos open parentheses 2 over 5 close parentheses plus 360 degree n comma end cell row x equals cell 293 comma 58 degree plus 360 degree n comma thin space end cell row blank blank blank row blank rightwards arrow cell x equals 180 degree plus 360 degree n end cell row x equals cell 180 degree plus 360 degree n end cell end table    

Dari penyelesaian diatas, yang memenuhi 0 degree less or equal than x less or equal than 360 degree adalah HP equals left curly bracket 66 comma 42182 degree semicolon space 180 degree semicolon space 293 comma 58 degree right curly bracket 

Jadi, himpunan penyelesaiannya adalah HP equals left curly bracket 66 comma 42182 degree semicolon space 180 degree semicolon space 293 comma 58 degree right curly bracket.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Yoga

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Hitunglah nilai eksak dari setiap bentuk dibawah ini  tan20∘tan40∘tan80∘

Pembahasan Soal:

Ingat tan(α)=cos(α)sin(α)

Sehingga diperoleh

 tan20tan40tan80=cos20cos40cos80sin20sin40sin80

  • Mencari nilai sin20sin40sin80

Ingat rumus perkalian trigonometri 
2sinαcosβ=sin(α+β)+sin(αβ).

Sehingga diperoleh 

======sin20sin40sin80sin20.21(2sin80sin40)sin20.21(cos(8040)cos(80+40))sin20.21(cos(40)cos(120))sin20.21(cos(40)(21))sin20.21(cos(40)+21)21sin20cos(40)+41

ingat 2sinαcosβ=sin(α+β)+sin(αβ)

21sin20cos(40)+41sin20=41(2sin20cos40)+41sin20=41(sin(20+40)+sin(2060))+41sin20=41(sin(60)+sin(20))+41sin20ingatsin(a)=sin(a)=41(213sin(20))+41sin20=81341sin(20)+41sin20=813

  • Mencari nilai cos20cos40cos80

Ingat rumus sudut rangkap trigonometri sin(2α)=2sin(a)cos(a).

Sehingga diperoleh 

=========(cos20cos40cos80)×2sin202sin202sin201(2sin20cos20)cos40cos802sin201sin40cos40cos802.2sin2012.sin40cos40cos804sin201sin80cos802.4sin2012.sin80cos808sin201sin1608sin201sin(18020)8sin201sin2081

Dari 2 perhitungan di atas kita dapatkan

tan20tan40tan80===cos20cos40cos80sin20sin40sin80818133 

Dengan demikian, nilai eksak dari tan20tan40tan80 adalah 3.

0

Roboguru

Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. e. 2cos(2x+30∘)cos(2x−60∘)=21​3​

Pembahasan Soal:

Ingat bahwa:

  • cos space left parenthesis a plus b right parenthesis equals cos space a space cos space b minus sin space a space sin space b cos space left parenthesis a minus b right parenthesis equals cos space a space cos space b plus sin space a space sin space b
  • cos squared open parentheses x close parentheses equals 1 minus sin squared open parentheses x close parentheses  

Mencari himpunan penyelesaian:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 cos open parentheses 2 x plus 30 to the power of ring operator thin space end exponent close parentheses cos open parentheses 2 x minus 60 to the power of ring operator thin space end exponent close parentheses end cell equals cell 1 half square root of 3 end cell row cell fraction numerator square root of 3 over denominator 2 end fraction cos squared open parentheses 2 x close parentheses plus sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin squared open parentheses 2 x close parentheses end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction end cell row cell fraction numerator square root of 3 over denominator 2 end fraction cos squared open parentheses 2 x close parentheses plus sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin squared open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction end cell equals 0 row cell fraction numerator square root of 3 cos squared open parentheses 2 x close parentheses minus square root of 3 sin squared open parentheses 2 x close parentheses minus square root of 3 plus 2 sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell equals 0 row cell fraction numerator square root of 3 cos squared open parentheses 2 x close parentheses minus square root of 3 sin squared open parentheses 2 x close parentheses minus square root of 3 plus 2 sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell equals 0 row cell 2 cos open parentheses 2 x close parentheses sin open parentheses 2 x close parentheses minus 2 sin squared open parentheses 2 x close parentheses square root of 3 end cell equals 0 row cell 2 sin open parentheses 2 x close parentheses open parentheses cos open parentheses 2 x close parentheses minus square root of 3 sin open parentheses 2 x close parentheses close parentheses end cell equals 0 end table end style  

sin open parentheses 2 x close parentheses equals 0 space atau space cos open parentheses 2 x close parentheses minus square root of 3 sin open parentheses 2 x close parentheses equals 0 x equals 180 degree n comma thin space x equals 90 degree plus 180 degree n comma thin space x equals 15 degree plus fraction numerator 180 degree n over denominator 2 end fraction  

Dari penyelesaian tersebut, yang memenuhi 0 degree less or equal than x less or equal than 360 degree adalah:

table attributes columnalign center left center end attributes row blank cell x equals 180 degree n end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 0 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 180 degree end cell blank row cell x equals 2 rightwards arrow end cell cell x equals 360 degree end cell blank row blank cell x equals 90 degree plus 180 degree n end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 90 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 270 degree end cell blank row blank cell x equals 15 degree plus fraction numerator 180 degree n over denominator 2 end fraction end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 15 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 105 degree end cell blank row cell x equals 2 rightwards arrow end cell cell x equals 195 degree end cell blank row cell x equals 3 rightwards arrow end cell cell x equals 285 degree end cell blank end table 

Jadi, himpunan penyelesaiannya adalah  HP equals left curly bracket 0 degree comma space 15 degree comma space 90 degree comma space 105 degree comma space 180 degree comma space 195 degree comma space 270 degree comma space 285 degree comma space 360 degree right curly bracket.

0

Roboguru

Jika sin(2x+60∘)=adan sin(x+45∘)=b, maka cos(3x+105∘)⋅cos(x+15∘)=…

Pembahasan Soal:

Ingat bahwa :

rumus sudut rangkap untuk cosinus adalah 

cos2A===cos2Asin2A12sin2A2cos2A1

Rumus perkalian Trigonometri

2cosAcosB=cos(A+B)+cos(AB)

dari soal diketahui 

sin(2x+60)=a

sin(x+45)=b

Maka cos(3x+105)cos(x+15) dapat ditentukan dengan cara berikut.

cos(3x+105)cos(x+15)=21(cos(3x+105+x+15)+cos(3x+105x+15))=21(cos(4x+120)+cos(2x+90))=21(cos2(2x+60)+cos2(x+45))=21(12sin2(2x+60)+12sin2(x+45))=21(22a22b2)=1a2b2

Jadi, jawaban yang tepat adalah B

0

Roboguru

Jika sin(2x+60∘)=a dan sin(x+45)=b maka sin(3x+105∘)⋅sin(x+15∘)=…

Pembahasan Soal:

Ingat bahwa :

rumus sudut rangkap untuk cosinus 

cos2A=cos2Asin2A

rumus perkalian Trigonometri

2sinAsinB==cos(A+B)cos(AB)21[cos(AB)cos(A+B)]

dari soal diketahui

sin(2x+60)=a

sin(x+45)=b

Maka 

sin(3x+105)sin(x+15)=21[cos(3x+105(x+15))cos(3x+105+x+15)]=21[cos(2x+90)cos(4x+120)]=21[cos2(x+45)cos2(2x+60)]=21[(1cos2(x+45)(1cos2(2x+60)))]=21[12b21+2a2]=21[2a22b2]=212(a2b2)=a2b2

Jadi, jawaban yang tepat adalah A

0

Roboguru

Gunakan rumus 2sinαsinβ=cos(α−β)−cos(α+β) untuk membuktikan: .

Pembahasan Soal:

Ingat bahwa:

  • cos space 2 a equals 1 minus 2 space sin squared a 
  • sin space a equals cos space left parenthesis 90 degree minus a right parenthesis 
  • sin space 2 a equals 2 space sin space a space cos space a 
  • 2 sin space alpha space sin space beta equals cos space left parenthesis alpha minus beta right parenthesis minus cos space left parenthesis alpha plus beta right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 54 degree space sin space 18 degree end cell equals cell 1 fourth end cell row cell cos space left parenthesis 90 degree minus 54 degree right parenthesis space sin space 18 degree end cell equals cell 1 fourth end cell row cell cos space 36 degree space sin space 18 degree end cell equals cell 1 fourth end cell end table 

Mencari sin space 18 degree:

Misalkan: theta equals 18 degree, maka:

 table attributes columnalign right center left columnspacing 0px end attributes row cell 5 theta end cell equals cell 90 degree end cell row cell 2 theta plus 3 theta end cell equals cell 90 degree end cell row cell 2 theta end cell equals cell 90 degree minus 3 theta end cell row cell sin 2 theta end cell equals cell sin left parenthesis 90 minus 3 theta right parenthesis equals cos 3 theta end cell row cell 2 sin theta cos theta end cell equals cell 4 cos cubed theta minus 3 cos theta end cell row cell 2 sin theta cos theta minus 4 cos cubed theta plus 3 cos theta end cell equals 0 row cell cos theta open parentheses 2 sin theta minus 4 cos squared theta plus 3 close parentheses end cell equals 0 row cell 2 sin theta minus 4 open parentheses 1 minus sin squared theta close parentheses plus 3 end cell equals 0 row cell sin theta end cell equals cell fraction numerator negative 2 plus-or-minus square root of 2 squared minus 4 cross times 4 cross times left parenthesis negative 1 right parenthesis end root over denominator 2 cross times 4 end fraction end cell row cell sin theta end cell equals cell fraction numerator negative 2 plus-or-minus square root of 20 over denominator 8 end fraction end cell row cell sin theta end cell equals cell fraction numerator square root of 5 minus 1 over denominator 4 end fraction end cell row cell sin space 18 degree end cell equals cell fraction numerator square root of 5 minus 1 over denominator 4 end fraction end cell end table 

Mencari cos space 36 degree:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 36 degree end cell equals cell cos space open parentheses 2 times 18 degree close parentheses end cell row blank equals cell 1 minus 2 sin squared 18 degree end cell row blank equals cell 1 minus 2 open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses squared end cell row blank equals cell 1 minus open parentheses fraction numerator square root of 5 plus 1 minus 2 square root of 5 over denominator 8 end fraction close parentheses end cell row blank equals cell fraction numerator square root of 5 plus 1 over denominator 4 end fraction end cell end table 

Sehingga didapat:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos open parentheses 36 degree close parentheses sin open parentheses 18 degree close parentheses end cell equals cell open parentheses fraction numerator square root of 5 plus 1 over denominator 4 end fraction close parentheses open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses square root of 5 plus 1 close parentheses open parentheses square root of 5 minus 1 close parentheses over denominator 4 times thin space 4 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus square root of 5 close parentheses open parentheses square root of 5 minus 1 close parentheses over denominator 16 end fraction end cell row blank equals cell 4 over 16 end cell row blank equals cell 1 fourth end cell end table 

Jadi, terbukti bahwa table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 54 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 18 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 fourth end cell end table.

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