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Nyatakan bentuk di bawah ini sebagai jumlah atau selisih kosinus 2sin200∘sin20∘

Pertanyaan

Nyatakan bentuk di bawah ini sebagai jumlah atau selisih kosinus

2sin200sin20

Pembahasan Soal:

Ingat rumus perkalian trigonometri 2sinαsinβ=cos(αβ)cos(α+β).

Sehingga diperoleh 

2sin200sin20==cos(20020)cos(200+20)cos(180)cos(220)

Dengan demikian, bentuk 2sin200sin20 sebagai jumlah atau selisih kosinus menjadi cos(180)cos(220).

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 12 September 2021

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Pertanyaan yang serupa

Hitunglah tanpa menggunakan kalkulator atau tabel trigonometri: a.

Pembahasan Soal:

Ingat definisi cosecan, c s c space x equals fraction numerator 1 over denominator sin space x end fraction maka

csc space 10 degree plus csc space 50 degree minus csc space 70 degree equals fraction numerator 1 over denominator sin space 10 degree end fraction plus fraction numerator 1 over denominator sin space 50 degree end fraction minus fraction numerator 1 over denominator sin space 70 degree end fraction equals fraction numerator sin space 50 degree space sin 70 degree plus sin space 10 degree space s in space 70 degree minus sin space 10 degree space sin space 50 degree over denominator sin space 10 degree space sin space 50 degree space sin space 70 degree end fraction equals fraction numerator sin 70 degree open parentheses sin space 50 degree plus sin space 10 degree close parentheses minus sin space 10 degree space sin space 50 degree over denominator sin space 10 degree space sin space 50 degree space sin space 70 degree end fraction equals A over B

Untuk A, gunakan rumus jumlah sinus dan perkalian sinus, ingat juga sudut berelasi.

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell sin 70 degree open parentheses sin space 50 degree plus sin space 10 degree close parentheses minus sin space 10 degree space sin space 50 degree end cell row blank equals cell sin space 70 degree open parentheses 2 space sin space 30 degree space cos space 20 degree close parentheses plus 1 half open parentheses cos space 60 degree minus cos space 40 degree close parentheses end cell row blank equals cell sin space 70 degree open parentheses 2 space times 1 half space sin space 70 degree close parentheses plus 1 half open parentheses 1 half minus cos space 40 degree close parentheses end cell row blank equals cell sin squared space 70 degree plus 1 fourth minus 1 half cos space 40 degree end cell row blank equals cell fraction numerator 1 minus cos space 2 open parentheses 70 degree close parentheses over denominator 2 end fraction plus 1 fourth minus 1 half cos space 40 degree end cell row blank equals cell 1 half plus 1 half cos space 40 degree plus 1 fourth minus 1 half cos space 40 degree end cell row blank equals cell 3 over 4 end cell end table

Untuk B, ingat rumus sudut rangkap pada sinus dan sudut berelasi.

table attributes columnalign right center left columnspacing 0px end attributes row B equals cell sin space 10 degree space sin space 50 degree space sin space 70 degree end cell row blank equals cell cos space 80 degree space cos space 40 degree space cos space 20 degree cross times fraction numerator 2 sin space 20 degree over denominator 2 sin space 20 degree end fraction end cell row blank equals cell fraction numerator cos space 80 degree space cos space 40 degree space sin space 40 degree over denominator 2 sin space 20 degree end fraction end cell row blank equals cell fraction numerator cos space 80 degree space sin space 80 degree over denominator 4 sin space 20 degree end fraction end cell row blank equals cell fraction numerator sin space 160 degree over denominator 8 sin space 20 degree end fraction end cell row blank equals cell fraction numerator sin space 20 degree over denominator 8 sin space 20 degree end fraction end cell row blank equals cell 1 over 8 end cell end table

dengan demikian diperoleh A over B equals fraction numerator begin display style bevelled 3 over 4 end style over denominator begin display style bevelled 1 over 8 end style end fraction equals 6.

Jadi, csc space 10 degree plus csc space 50 degree minus csc space 70 degree equals 6.

Roboguru

Tanpa menggunakan kalkulator atau tabel trigonometri, hitunglah:

Pembahasan Soal:

Ingat kembali:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space sin space straight alpha space sin space straight beta end cell equals cell negative cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses end cell row cell 2 space sin space straight alpha space cos space straight beta end cell equals cell sin space open parentheses straight alpha plus straight beta close parentheses plus sin space open parentheses straight alpha minus straight beta close parentheses end cell row cell sin space open parentheses 180 degree minus straight A close parentheses end cell equals cell sin space straight A end cell end table 

Maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 8 space sin space 70 degree times sin space 50 degree times sin space 10 degree end cell equals cell 4 times 2 times sin space 70 degree times sin space 50 degree times sin space 10 degree end cell row blank equals cell open parentheses 4 times sin space 70 degree close parentheses times open parentheses 2 times sin space 50 degree times sin space 10 degree close parentheses end cell row blank equals cell open parentheses 4 times sin space 70 degree close parentheses times open parentheses negative cos space open parentheses 50 degree plus 10 degree close parentheses plus cos space open parentheses 50 degree minus 10 degree close parentheses close parentheses end cell row blank equals cell open parentheses 4 times sin space 70 degree close parentheses times open parentheses negative cos space 60 degree plus cos space 40 degree close parentheses end cell row blank equals cell open parentheses 4 times sin space 70 degree close parentheses times open parentheses negative 1 half plus cos space 40 degree close parentheses end cell row blank equals cell negative 2 space sin space 70 degree plus 4 times sin space 70 degree times cos space 40 degree end cell row blank equals cell negative 2 space sin space 70 degree plus 2 times open parentheses 2 times sin space 70 degree times cos space 40 degree close parentheses end cell row blank equals cell negative 2 space sin space 70 degree plus 2 times open parentheses sin space open parentheses 70 degree plus 40 degree close parentheses plus sin space open parentheses 70 degree minus 40 degree close parentheses close parentheses end cell row blank equals cell negative 2 space sin space 70 degree plus 2 times open parentheses sin space 110 degree plus sin space 30 degree close parentheses end cell row blank equals cell negative 2 space sin space 70 degree plus 2 times open parentheses sin space 110 degree plus 1 half close parentheses end cell row blank equals cell negative 2 space sin space 70 degree plus 2 space sin space 110 degree plus 1 end cell row blank equals cell negative 2 space sin space 70 degree plus 2 space sin space open parentheses 180 degree minus 70 degree close parentheses plus 1 end cell row blank equals cell up diagonal strike negative 2 space sin space 70 degree end strike up diagonal strike plus 2 space sin space 70 degree end strike plus 1 end cell row blank equals 1 end table end style 

Jadi, nilai table attributes columnalign right center left columnspacing 0px end attributes row blank blank 8 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 70 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 times 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 50 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 times 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 10 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 1 end table.

Roboguru

Nilai dari

Pembahasan Soal:

Ingat kembali rumus-rumus pengurangan sinus dan penjumlahan cosinus berikut.

  1. sin space alpha minus sin space beta equals 2 space cos space open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses space sin space open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses
  2. cos space alpha plus cos space beta equals 2 space cos space open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses

Begitu juga dengan rumus perkalian sinus dan kosinus:

  1. 2 space sin space alpha space sin space beta equals cos space open parentheses alpha minus beta close parentheses minus cos space open parentheses alpha plus beta close parentheses
  2. 2 space cos space alpha space cos space beta equals cos space open parentheses alpha plus beta close parentheses plus cos space open parentheses alpha minus beta close parentheses

Berdasarkan rumus-rumus di atas, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 75 degree open parentheses sin space 55 degree minus sin space 25 degree close parentheses over denominator cos space 55 degree plus cos space 25 degree end fraction end cell equals cell tan space 75 degree fraction numerator up diagonal strike 2 space cos space open parentheses begin display style fraction numerator 55 degree plus 25 degree over denominator 2 end fraction end style close parentheses end strike space sin space open parentheses begin display style fraction numerator 55 degree minus 25 degree over denominator 2 end fraction end style close parentheses over denominator up diagonal strike 2 space cos space open parentheses begin display style fraction numerator 55 degree plus 25 degree over denominator 2 end fraction end style close parentheses end strike space cos space open parentheses begin display style fraction numerator 55 degree minus 25 degree over denominator 2 end fraction end style close parentheses end fraction end cell row blank equals cell tan space 75 degree fraction numerator sin space 15 degree over denominator cos space 15 degree end fraction end cell row blank equals cell fraction numerator sin space 75 degree space sin space 15 degree over denominator cos space 75 degree space cos space 15 degree end fraction end cell row blank equals cell fraction numerator up diagonal strike begin display style 1 half end style end strike open parentheses cos space 60 degree minus cos space 90 degree close parentheses over denominator up diagonal strike begin display style 1 half end style end strike open parentheses cos space 90 degree plus cos space 60 degree close parentheses end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style minus 0 over denominator 0 plus begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style bevelled 1 half end style over denominator begin display style bevelled 1 half end style end fraction end cell row blank equals 1 end table 

Dengan demikian, hasil dari fraction numerator tan space 75 degree open parentheses sin space 55 degree minus sin space 25 degree close parentheses over denominator cos space 55 degree plus cos space 25 degree end fraction adalah 1.

Oleh karena itu, jawaban yang benar adalah E.

Roboguru

Hitunglah tanpa menggunakan tabel trigonometri maupun kalkulator. sin6∘sin42∘sin66∘sin78∘

Pembahasan Soal:

Ingat rumus perkalian trigonometri
2sinαsinβ=cos(αβ)cos(α+β)

Sehingga diperoleh perhitungan 

sin42sin78sin6sin66=================21(2sin42sin78)21(cos(4278)cos(42+78))21(cos(36)cos(120))ingatcos(α)=cosα21(cos36(21))21(cos36+21) ingatcos36=45+121(45+1+21)21(45+1+42)2×45+385+321(2sin6sin66)21(cos(666)cos(6+66))21(cos(60)cos72)ingatcos(α)=cosα21(21cos72) ingatcos72=45121(21451)21(42451)2×435835

sehingga diperoleh 

sin6sin42sin66sin78======83+5×8358(3+5)×8(35)6432(5)26495644161

Dengan demikian, hasil dari sin6sin42sin66sin78 adalah 161

Roboguru

Tunjukkan bahwa: sin10∘sin30∘sin50∘sin70∘=161​

Pembahasan Soal:

Ingat rumus perkalian trigonometri 
2sinαsinβ=cos(αβ)cos(α+β).

Sehingga diperoleh 

=======sin10sin30sin50sin7021sin10sin50sin7021.21(2sin10sin50)41(cos(1050)cos(10+50))sin7041(cos(40)cos(60))sin7041(cos(40)21)sin70(41cos(40)81)sin7041cos40sin7081sin70

ingat 2cosαsinβ=sin(α+β)sin(αβ)

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 fourth end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 40 degree end cell 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 70 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 over 8 end cell 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals 1 over 8 open parentheses 2 space cos table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 40 degree end cell 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table close parentheses plus degree table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 over 8 end cell 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals 1 over 8 open parentheses sin open parentheses 40 degree plus 70 degree close parentheses minus sin open parentheses 40 degree minus 70 degree close parentheses close parentheses minus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 over 8 end cell 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals 1 over 8 open parentheses sin open parentheses 110 degree close parentheses minus sin open parentheses negative 30 degree close parentheses close parentheses minus 1 over 8 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table space text ingat end text space sin open parentheses negative a close parentheses equals negative sin open parentheses a close parentheses equals 1 over 8 open parentheses sin open parentheses 110 degree close parentheses minus open parentheses negative sin open parentheses 30 degree close parentheses close parentheses close parentheses minus 1 over 8 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 70 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals 1 over 8 open parentheses sin open parentheses 110 degree close parentheses plus 1 half close parentheses minus 1 over 8 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 70 end table degree equals 1 over 8 sin open parentheses 180 degree minus 70 degree close parentheses plus 1 over 16 minus 1 over 8 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 70 end table degree equals 1 over 8 sin open parentheses 70 degree close parentheses plus 1 over 16 minus 1 over 8 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 70 end table degree equals 1 over 16

Dengan demikian, terbukti bahwa sin10sin30sin50sin70=161.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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