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Tunjukkan bahwa: sin 2 0 ∘ sin 4 0 ∘ sin 6 0 ∘ sin 8 0 ∘ = 16 3 ​

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S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Jawaban

terbukti bahwa sin 2 0 ∘ sin 4 0 ∘ sin 6 0 ∘ sin 8 0 ∘ = 16 3 ​ .

 terbukti bahwa .

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Pembahasan

Ingat rumus perkalian trigonometri 2 sin α sin β = cos ( α − β ) − cos ( α + β ) . Sehingga diperoleh ​ = = = = = = ​ sin 2 0 ∘ sin 4 0 ∘ sin 6 0 ∘ sin 8 0 ∘ sin 2 0 ∘ . 2 1 ​ ( 2 sin 8 0 ∘ sin 4 0 ∘ ) sin 6 0 ∘ sin 2 0 ∘ . 2 1 ​ ( cos ( 8 0 ∘ − 4 0 ∘ ) − cos ( 8 0 ∘ + 4 0 ∘ ) ) sin 6 0 ∘ sin 2 0 ∘ . 2 1 ​ ( cos ( 4 0 ∘ ) − cos ( 12 0 ∘ ) ) sin 6 0 ∘ sin 2 0 ∘ . 2 1 ​ ( cos ( 4 0 ∘ ) − ( − 2 1 ​ ) ) sin 6 0 ∘ sin 2 0 ∘ . 2 1 ​ ( cos ( 4 0 ∘ ) + 2 1 ​ ) sin 6 0 ∘ ( 2 1 ​ sin 2 0 ∘ cos ( 4 0 ∘ ) + 4 1 ​ ) sin 6 0 ∘ ​ ingat 2 sin α cos β = sin ( α + β ) + sin ( α − β ) Dengan demikian,terbukti bahwa sin 2 0 ∘ sin 4 0 ∘ sin 6 0 ∘ sin 8 0 ∘ = 16 3 ​ .

Ingat rumus perkalian trigonometri 
.

Sehingga diperoleh 

ingat 

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 .

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