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Nyatakan bentuk dibawah ini sebagai jumlah atau selisih kosinus sin(2a+b)sin(2a−b)

Pertanyaan

Nyatakan bentuk dibawah ini sebagai jumlah atau selisih kosinus

sin(2a+b)sin(2ab)

Pembahasan Soal:

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

Sehingga diperoleh 

====sin(2a+b)sin(2ab)21(2sin(2a+b)sin(2ab))21(cos((2a+b)(2ab))cos((2a+b)+(2ab)))21(cos2bcos4a)21cos2b21cos4a

Dengan demikian, bentuk sin(2a+b)sin(2ab) sebagai jumlah atau selisih kosinus menjadi 21cos2b21cos4a.

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

Tunjukkan bahwa .

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 cos space straight alpha space sin space straight beta end cell equals cell sin space open parentheses straight alpha plus straight beta close parentheses minus sin space open parentheses straight alpha minus straight beta close parentheses end cell end table 

Akan dibuktikan 8 space sin space 20 degree times sin space 40 degree times sin space 80 degree equals square root of 3. Maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell 8 space sin space 20 degree times sin space 40 degree times sin space 80 degree end cell row blank equals cell 4 times 2 times sin space 20 degree times sin space 40 degree times sin space 80 degree end cell row blank equals cell open parentheses 4 space sin space 20 degree close parentheses open parentheses 2 times sin space 80 degree times sin space 40 degree close parentheses end cell row blank equals cell open parentheses 4 space sin space 20 degree close parentheses open parentheses negative cos space open parentheses 80 degree plus 40 degree close parentheses plus cos space open parentheses 80 degree minus 40 degree close parentheses close parentheses end cell row blank equals cell open parentheses 4 space sin space 20 degree close parentheses open parentheses negative cos space 120 degree plus cos space 40 degree close parentheses end cell row blank equals cell open parentheses 4 space sin space 20 degree close parentheses open parentheses negative open parentheses negative 1 half close parentheses plus cos space 40 degree close parentheses end cell row blank equals cell open parentheses 4 space sin space 20 degree close parentheses open parentheses 1 half plus cos space 40 degree close parentheses end cell row blank equals cell 2 space sin space 20 degree plus 4 times sin space 20 degree times cos space 40 degree end cell row blank equals cell 2 space sin space 20 degree plus 2 times 2 times sin space 20 degree times cos space 40 degree end cell row blank equals cell 2 space sin space 20 degree plus 2 times open parentheses 2 times cos space 40 degree times sin space 20 degree close parentheses end cell row blank equals cell 2 space sin space 20 degree plus 2 open parentheses sin space open parentheses 40 degree plus 20 degree close parentheses minus sin space open parentheses 40 degree minus 20 degree close parentheses close parentheses end cell row blank equals cell 2 space sin space 20 degree plus 2 open parentheses sin space 60 degree minus sin space 20 degree close parentheses end cell row blank equals cell 2 space sin space 20 degree plus 2 open parentheses 1 half square root of 3 minus sin space 20 degree close parentheses end cell row blank equals cell up diagonal strike 2 space sin space 20 degree end strike plus square root of 3 up diagonal strike negative 2 space sin space 20 degree end strike end cell row blank equals cell square root of 3 end cell end table end style 

Jadi, terbukti bahwa 8 space sin space 20 degree times sin space 40 degree times sin space 80 degree equals square root of 3.

Roboguru

Tentukan himpunan penyelesaian tiap persamaan berikut untuk . g.

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.

Roboguru

Buktikan bahwa: sin3θsinθ+sin6θsin2θ+sin12θsin4θ+sin24θsin8θ+sin48θsin16θ+sin96θsin32θ=sin65θsin63θ

Pembahasan Soal:

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

Diperoleh perhitungan sebagai  berikut

sin3θsinθsin6θsin2θsin12θsin4θsin24θsin8θsin48θsin16θsin96θsin32θ========================21(2sin3θsinθ)21(cos(3θθ)cos(3θ+θ))21(cos2θcos4θ)21cos2θ21cos4θ21(2sin6θsin2θ)21(cos(6θ2θ)cos(6θ+2θ))21(cos4θcos8θ)21cos4θ21cos8θ21(2sin12θsin4θ)21(cos(12θ4θ)cos(12θ+4θ))21(cos8θcos16θ)21cos8θ21cos16θ21(2sin24θsin8θ)21(cos(24θ8θ)cos(24θ+8θ))21(cos16θcos32θ)21cos16θ21cos32θ21(2sin48θsin16θ)21(cos(48θ16θ)cos(48θ+16θ))21(cos32θcos64θ)21cos32θ21cos64θ21(2sin96θsin32θ)21(cos(96θ32θ)cos(96θ+32θ))21(cos64θcos128θ)21cos64θ21cos128θ

Sehingga diperoleh

sin3θsinθ+sin6θsin2θ+sin12θsin4θ+sin24θsin8θ+sin48θsin16θ+sin96θsin32θ=21cos2θ21cos4θ+21cos4θ21cos8θ+21cos8θ21cos16θ+21cos16θ21cos32θ+21cos32θ21cos64θ+21cos64θ21cos128θ=21cos2θ21cos128θ=21(cos2θcos128θ)

Ingat rumus jumlah dan selisih trigonometri 
cosαcosβ=2sin21(α+β)sin21(αβ).

======21(cos2θcos128θ)21(2sin21(2θ+128θ)sin21(2θ128θ))21(2sin21(130θ)sin21(126θ))21(2sin65θsin(63θ))sin65θsin(63θ)sin65θsin63θsin65θsin63θ

Dengan demikian, terbukti bahwa sin3θsinθ+sin6θsin2θ+sin12θsin4θ+sin24θsin8θ+sin48θsin16θ+sin96θsin32θ=sin65θsin63θ

Roboguru

Nilai eksak dari ekspresi trigonometri: tan5∘⋅tan25∘⋅tan45∘⋅tan65∘⋅tan85∘ adalah . . . .

Pembahasan Soal:

Ingat identitas trigonometri tanx=cosxsinx dan rumus perkalian trigonometri berikut ini:

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

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

Dengan menggunakan konsep tersebut, diperoleh hasil:

tan space 5 degree times tan space 25 degree times tan space 45 degree times tan space 65 degree times tan space 85 degree equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times tan space 45 degree times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times 1 times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals open parentheses fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction times fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction close parentheses open parentheses fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction close parentheses equals open parentheses fraction numerator sin space 85 degree space sin space 5 degree over denominator cos space 85 degree space cos space 5 degree end fraction close parentheses open parentheses fraction numerator sin space 65 degree space sin space 25 degree over denominator cos space 65 degree space cos space 25 degree end fraction close parentheses equals open parentheses fraction numerator 1 half open parentheses negative cos space open parentheses 85 degree plus 5 degree close parentheses plus cos space open parentheses 85 degree minus 5 degree close parentheses close parentheses over denominator 1 half open parentheses cos space open parentheses 85 degree plus 5 degree close parentheses plus cos space open parentheses 85 degree minus 5 degree close parentheses close parentheses end fraction close parentheses open parentheses fraction numerator 1 half open parentheses negative cos space open parentheses 65 degree plus 25 degree close parentheses plus cos space open parentheses 65 degree minus 25 degree close parentheses close parentheses over denominator 1 half open parentheses cos space open parentheses 65 degree plus 25 degree close parentheses plus cos space open parentheses 65 degree minus 25 degree close parentheses close parentheses end fraction close parentheses equals open parentheses fraction numerator 1 half open parentheses negative cos space 90 degree plus cos space 80 degree close parentheses over denominator 1 half open parentheses cos space 90 degree plus cos space 80 degree close parentheses end fraction close parentheses open parentheses fraction numerator 1 half open parentheses negative cos space 90 degree plus cos space 40 degree close parentheses over denominator 1 half open parentheses cos space 90 degree plus cos space 40 degree close parentheses end fraction close parentheses equals open parentheses fraction numerator negative cos space 90 degree plus cos space 80 degree over denominator cos space 90 degree plus cos space 80 degree end fraction close parentheses open parentheses fraction numerator negative cos space 90 degree plus cos space 40 degree over denominator cos space 90 degree plus cos space 40 degree end fraction close parentheses equals open parentheses fraction numerator 0 plus cos space 80 degree over denominator 0 plus cos space 80 degree end fraction close parentheses open parentheses fraction numerator 0 plus cos space 40 degree over denominator 0 plus cos space 40 degree end fraction close parentheses equals open parentheses fraction numerator cos space 80 degree over denominator cos space 80 degree end fraction close parentheses open parentheses fraction numerator cos space 40 degree over denominator cos space 40 degree end fraction close parentheses equals 1 times 1 equals 1

Jadi, tan5tan25tan45tan65tan85=1.

Oleh karena itu, jawaban yang benar adalah E.

Roboguru

Hitunglah tanpa menggunakan tabel trigonometri maupun kalkulator. sin12∘sin48∘sin54∘

Pembahasan Soal:

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

Sehingga diperoleh perhitungan sebagai berikut

sin12sin48sin54=21(2sin12sin48)sin54=21(cos(1248)cos(12+48))sin54=21(cos(36)cos(60))sin54 ingatcos(α)=cosα=21(cos3621)sin54ingatsinα=cos(90a)=21(cos3621)cos(9054)=21(cos3621)cos36ingat cos36°=45+1=21(45+121)45+1=21(45+142)45+1=2×4×4(51)(5+1) ingat(ab)(a+b)=a2b2=32(5)212=3251=324=81

Dengan demikian, hasil dari sin12sin48sin54 adalah 81.

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