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Tanpa menggunakan kalkulator atau tabel trigonometri, hitunglah: 8sin70∘⋅sin50∘⋅sin10∘

Pertanyaan

Tanpa menggunakan kalkulator atau tabel trigonometri, hitunglah:

8 space sin space 70 degree times sin space 50 degree times sin space 10 degree 

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Lestari

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Hitunglah nilai eksak dari setiap bentuk di bawah ini  sin20∘sin40∘sin80∘

Pembahasan Soal:

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

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

Dengan demikian, nilai eksak dari sin20sin40sin80 adalah 813.

0

Roboguru

Tunjukkan bahwa: sin20∘sin40∘sin60∘sin80∘=163​

Pembahasan Soal:

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

Sehingga diperoleh 

======sin20sin40sin60sin80sin20.21(2sin80sin40)sin60sin20.21(cos(8040)cos(80+40))sin60sin20.21(cos(40)cos(120))sin60sin20.21(cos(40)(21))sin60sin20.21(cos(40)+21)sin60(21sin20cos(40)+41)sin60

ingat 2sinαcosβ=sin(α+β)+sin(αβ)

open parentheses 1 half sin 20 degree cos open parentheses 40 degree close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals open parentheses 1 fourth open parentheses 2 space sin 20 degree space cos 40 degree close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals open parentheses 1 fourth open parentheses sin open parentheses 20 degree plus 40 degree close parentheses plus sin open parentheses 20 degree minus 60 degree close parentheses close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals open parentheses 1 fourth open parentheses sin open parentheses 60 degree close parentheses plus sin open parentheses negative 20 degree close parentheses close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 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 open parentheses 1 fourth open parentheses 1 half square root of 3 minus sin open parentheses 20 degree close parentheses close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals open parentheses 1 over 8 square root of 3 minus 1 fourth sin open parentheses 20 degree close parentheses plus 1 fourth sin 20 degree close parentheses space 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 60 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table equals 1 over 8 square root of 3 space. 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 60 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 equals blank end table 1 over 8 square root of 3 space.1 half square root of 3 space equals 3 over 16

Dengan demikian, terbukti bahwa sin20sin40sin60sin80=163.

0

Roboguru

Nilai eksak dari (sin10∘sin50∘sin70∘) adalah . . . .

Pembahasan Soal:

Ingat kembali bahwa:

2sinAcosB=sin(A+B)+sin(AB)2sinAsinB=(cos(A+B)cos(AB))sinA+sinB=cos(2A+B)sin(2AB)  

Sehingga diperoleh perhitungan:

================sin10sin50sin70(sin10sin50)sin70(21[2sin10sin50])sin70(21[cos(50+10)cos(5010)])sin7021sin70(cos60cos40)21sin70(21cos40)41sin70+21sin70cos4041sin70+41(2sin70cos40)41sin70+41(sin(70+40)+sin(7040)41sin70+41(sin110+sin30)41sin70+41(sin110+21)41sin70+41sin110+8141(sin110sin70)+8141(cos2110+70sin211070)+8141(cos90sin20)+8141(0sin20)+8181   

Dengan demikian, nilai dari (sin10sin50sin70) adalah 81

Jadi jawaban yang tepat adalah A.

0

Roboguru

Tentukanlah nilai dari: sin10∘sin50∘sin70∘

Pembahasan Soal:

Konsep:

2 space sin space A space sin space B equals cos space left parenthesis A minus B right parenthesis minus space cos space left parenthesis A plus B right parenthesis 2 space sin space A space cos space B equals sin space left parenthesis A equals B right parenthesis plus sin space left parenthesis A minus B right parenthesis

Pembahasan:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin space 10 degree space sin space 50 degree space sin space 70 degree end cell row blank equals cell negative 1 half open parentheses negative 2 space sin space 50 degree space sin space 10 degree close parentheses sin space 70 degree end cell row blank equals cell negative 1 half sin space 70 degree open parentheses negative 2 space sin space 50 degree space sin space 10 degree close parentheses end cell row blank equals cell negative 1 half sin space 70 degree open parentheses open parentheses cos space left parenthesis 50 degree plus 10 degree close parentheses minus cos open parentheses 50 degree minus 10 degree close parentheses close parentheses end cell row blank equals cell negative 1 half space sin space 70 degree open parentheses cos space 60 degree minus cos space 40 degree close parentheses end cell row blank equals cell negative 1 half space sin space 70 degree open parentheses 1 half minus cos space 40 degree close parentheses end cell row blank equals cell negative 1 fourth space sin space 70 degree plus 1 half.1 half open parentheses 2 space sin space 70 degree cos space 40 degree close parentheses end cell row blank equals cell negative 1 fourth space sin space 70 degree plus 1 fourth open parentheses sin open parentheses 70 degree plus 40 degree close parentheses plus sin open parentheses 70 degree minus 40 degree close parentheses close parentheses end cell row blank equals cell negative 1 fourth space sin space 70 degree plus 1 fourth open parentheses sin space 110 degree plus sin space 30 degree close parentheses end cell row blank equals cell negative 1 fourth space sin space 70 degree plus 1 fourth open parentheses sin space 110 degree plus 1 half close parentheses end cell row blank equals cell negative 1 fourth space sin space 70 degree plus 1 fourth sin space 110 degree plus 1 over 8 end cell row blank equals cell 1 over 8 end cell end table

Jadi, nilai dari sin space 10 degree space sin space 50 degree space sin space 70 degree equals 1 over 8.

1

Roboguru

Perhatikan pernyataan berikut. i) 2sin40∘cos20∘ii) 2cos50∘sin70∘iii) 2sin70∘sin10∘iv) 2cos80∘cos20∘​====​sin30∘+sin20∘sin120∘+sin20∘−cos80∘+cos60∘cos100∘+cos30∘​ Pernyataan yang benar adalah ....

Pembahasan Soal:

Ingat rumus perkalian trigonometri

2sinαcosβ2cosαsinβ2cosαcosβ2sinαsinβ====sin(α+β)+sin(αβ)sin(α+β)sin(αβ)cos(α+β)+cos(αβ)cos(α+β)cos(αβ)

Sehingga diperoleh

i) 2sin40cos20ii) 2cos50sin70iii) 2sin70sin10iv) 2cos80cos20===========sin(40+20)+sin(4020)sin60+sin20sin(50+70)sin(5070)sin120sin(20)ingatsin(α)=sinαsin120+sin20(2sin70sin10)(cos(70+10)cos(7010))(cos80cos60)cos80+cos60cos(80+20)cos(8020)cos100cos60

Dengan demikian, jawaban yang benar adalah D.

 

0

Roboguru

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