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Tunjukanlah bahwa pernyataan-pernyataan berikut adalah benar.

Pertanyaan

Tunjukanlah bahwa pernyataan-pernyataan berikut adalah benar.

sin space 20 degree space sin space 40 degree space sin space 80 degree equals 1 over 8 square root of 3

Pembahasan Soal:

Membuktian pernyataan tersebut dengan mengaplikasikan rumus-rumus selisih sinus dan perkalian trigonometri:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space A minus sin space B end cell equals cell 2 space cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin space open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses end cell row cell negative 2 space sin space A space sin space B end cell equals cell cos space left parenthesis A plus B right parenthesis minus cos space left parenthesis A minus B right parenthesis end cell end table

Membuktikan pernyataan di bawah ini benar:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin space 20 degree space sin space 40 degree close parentheses space sin space 80 degree end cell row blank equals cell open parentheses sin space 40 degree space sin space 20 degree close parentheses space sin space 80 degree end cell row blank equals cell negative 1 half open parentheses cos open parentheses 40 degree plus 20 degree close parentheses minus cos space open parentheses 40 degree minus 20 degree close parentheses close parentheses sin space 80 degree end cell row blank equals cell negative 1 half open parentheses cos space 60 degree minus cos space 20 degree close parentheses space sin space 80 degree end cell row blank equals cell negative 1 half open parentheses 1 half minus cos space 20 degree close parentheses sin space 80 degree end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 half sin space 80 degree cos space 20 degree end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 half open parentheses sin space 80 degree cos space 20 degree close parentheses end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 half.1 half open parentheses sin space left parenthesis 80 degree plus 20 degree right parenthesis plus sin space open parentheses 80 degree minus 20 degree close parentheses close parentheses end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 fourth open parentheses sin space 100 degree plus sin space 60 degree close parentheses end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 fourth open parentheses sin space 100 degree plus 1 half square root of 3 close parentheses end cell row blank equals cell negative 1 fourth sin space 80 degree plus 1 fourth sin space 100 degree plus 1 over 8 square root of 3 end cell row blank equals cell negative 1 fourth open parentheses sin space 80 degree minus sin space 100 degree close parentheses plus 1 over 8 square root of 3 end cell row blank equals cell negative 1 fourth open parentheses 2 space cos open parentheses fraction numerator 80 degree plus 100 degree over denominator 2 end fraction close parentheses sin space open parentheses fraction numerator 80 degree minus 100 degree over denominator 2 end fraction close parentheses close parentheses plus 1 over 8 square root of 3 end cell row blank equals cell negative 1 fourth open parentheses 2 space cos space 90 degree sin space left parenthesis negative 10 degree close parentheses right parenthesis plus 1 over 8 square root of 3 end cell row blank equals cell negative 1 fourth open parentheses 2 left parenthesis 0 right parenthesis left parenthesis sin space left parenthesis negative 10 degree right parenthesis close parentheses plus 1 over 8 square root of 3 end cell row blank equals cell 0 plus 1 over 8 square root of 3 end cell row blank equals cell 1 over 8 square root of 3 end cell end table

Terbukti bahwa sin space 20 degree space sin space 40 degree space sin space 80 degree equals 1 over 8 square root of 3 adalah benar.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

O. Rahmawati

Mahasiswa/Alumni UIN Sunan Gunung Djati Bandung

Terakhir diupdate 11 Juli 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

sin24xcos2x=....

Pembahasan Soal:

Ingat rumus perkalian trigonometri 
2sinαcosβ2sinαsinβ==sin(α+β)+sin(αβ)cos(αβ)cos(α+β).

Sehingga diperoleh 

sin24xcos2x=============sin4xsin4xcos2xsin4x21(2sin4xcos2x)21sin4x(sin(4x+2x)+sin(4x2x))21sin4x(sin6x+sin2x)21(sin4x.sin6x)+21(sin4x.sin2x)41(2sin4x.sin6x)+41(2sin4x.sin2x)41(cos(4x6x)cos(4x+6x))+41(cos(4x2x)cos(4x+2x))41(cos(2x)cos10x)+41(cos2xcos6x)ingat  cos(α)=cos(α)41(cos2xcos10x)+41(cos2xcos6x)41cos2x41cos10x+41cos2x41cos6x21cos2x41cos10x41cos6x41(2cos2xcos10xcos6x)41(2cos2xcos6xcos10x)

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

tan20⋅tan40⋅tan80=....

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)

2cosAsinB=sin(A+B)sin(AB)

Dengan menggunakan konsep tersebut, diperoleh hasil:

tan space 20 degree space tan space 40 degree space tan space 80 degree equals fraction numerator sin space 20 degree over denominator cos space 20 degree end fraction times fraction numerator sin space 40 degree over denominator cos space 40 degree end fraction times fraction numerator sin space 80 degree over denominator cos space 80 degree end fraction equals fraction numerator sin space 20 degree blank open parentheses sin space 80 degree blank sin space 40 degree close parentheses over denominator cos space 20 degree open parentheses cos space 80 degree cos space 40 degree close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half 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 close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses 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 close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half open parentheses negative cos blank 120 degree plus cos space 40 degree close parentheses close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses cos space 120 degree plus cos space 40 degree close parentheses close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half open parentheses 1 half plus cos space 40 degree close parentheses close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses negative 1 half plus cos space 40 degree close parentheses close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 fourth plus 1 half blank cos space 40 degree close parentheses over denominator cos space 20 degree open parentheses negative 1 fourth plus 1 half blank cos space 40 degree close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank cos space 40 degree space sin space 20 degree over denominator negative 1 fourth cos space 20 degree plus 1 half blank cos space 40 degree cos space 20 degree end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank open parentheses 1 half 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 close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 half blank open parentheses 1 half open parentheses cos space open parentheses 40 degree plus 20 degree close parentheses plus cos space open parentheses 40 degree minus 20 degree close parentheses close parentheses close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank open parentheses 1 half open parentheses sin space 60 degree minus sin space 20 degree close parentheses close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 half blank open parentheses 1 half open parentheses cos space 60 degree minus cos space 20 degree close parentheses close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 fourth blank open parentheses sin space 60 degree minus sin space 20 degree close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 fourth blank open parentheses cos space 60 degree plus cos space 20 degree close parentheses end fraction equals fraction numerator sin space 20 degree plus open parentheses sin space 60 degree minus sin space 20 degree close parentheses over denominator negative cos space 20 degree plus open parentheses cos space 60 degree plus cos space 20 degree close parentheses end fraction equals fraction numerator sin space 60 degree over denominator cos space 60 degree end fraction equals tan space 60 degree equals square root of 3

Jadi, tan20tan40tan80=3.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Tunjukkan bahwa: sin36∘sin72∘sin108∘sin144∘=165​

Pembahasan Soal:

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

Sehingga diperoleh perhitungan 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell size 12px sin size 12px 36 size 12px degree size 12px space size 12px sin size 12px 72 size 12px degree size 12px space size 12px sin size 12px 108 size 12px degree size 12px space size 12px sin size 12px 144 size 12px degree end cell row blank size 12px equals cell size 12px space size 12px sin size 12px 36 size 12px degree size 12px space size 12px sin size 12px 72 size 12px degree size 12px space size 12px sin begin mathsize 12px style left parenthesis 180 degree minus 72 degree right parenthesis end style size 12px space size 12px sin size 12px left parenthesis size 12px 180 size 12px degree size 12px minus size 12px 36 size 12px degree size 12px right parenthesis size 12px space end cell row blank size 12px equals cell size 12px space size 12px sin size 12px 36 size 12px degree size 12px space size 12px sin size 12px 72 size 12px degree size 12px sin size 12px 72 size 12px degree size 12px sin size 12px 36 size 12px degree size 12px space end cell row blank size 12px equals cell begin mathsize 12px style left parenthesis sin 36 degree space sin 72 degree right parenthesis end style to the power of size 12px 2 end cell end table

Error converting from MathML to accessible text.

Dengan demikian, terbukti bahwa sin36sin72sin108sin144=165.

0

Roboguru

Tunjukkan bahwa: tan6∘tan42∘tan66∘tan78∘=1

Pembahasan Soal:

Ingat rumus perkalian trigonometri
2sinαsinβ2cosαcosβ==cos(αβ)cos(α+β)cos(α+β)+cos(αβ)

ingat tanα=cosαsinα, sehingga di dapat 

tan6tan42tan66tan78=cos6cos42cos66cos78sin6sin42sin66sin78

Mencari sin6sin42sin66sin78

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

Mencari cos6cos42cos66cos78

cos6cos42cos66cos78=21(2cos6cos66)21(2cos42cos78)=41(cos(6+66)+cos(666))(cos(42+78)+cos(4278))=41(cos72+cos(60))(cos120+cos(36))ingat cos(α)=cosα=41(cos72+cos60)(cos120+cos36)=41(cos72+21)(21+cos36)=41(21cos72+cos72cos3641+21cos36)=81(cos72+2cos72cos3621+cos36)=81(cos72+(cos(72+36)+cos(7236))21+cos36)=81(cos72+cos108+cos3621+cos36)ingatcos(180α)=cosα=81(cos72+cos(18072)+2cos3621)=81(cos72cos72+2cos3621)=81(2cos72+2cos3621)=81(2cos72+2cos36)×cos18cos18161=81cos18(2cos72cos18+2cos36cos18)161=161+81cos18((cos(72+18)+cos(7218)))+cos18(cos(36+18)+cos(3618))=161+81cos18((cos90+cos54)+(cos54+cos18))=161+81cos18(0cos54+cos54+cos18)=161+81cos18cos18=161+81=161+2=161

Sehingga didapat 

tan6tan42tan66tan78===cos6cos42cos66cos78sin6sin42sin66sin781611611

Dengan demikian, terbukti bahwa tan6tan42tan66tan78=1.

0

Roboguru

Buktikan identitas berikut. d.

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 end table 

Akan dibuktikan sin space open parentheses straight pi over 4 plus straight theta close parentheses times sin space open parentheses straight pi over 4 minus straight theta close parentheses equals 1 half space cos space 2 straight theta. Maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell sin space open parentheses straight pi over 4 plus straight theta close parentheses times sin space open parentheses straight pi over 4 minus straight theta close parentheses end cell row blank equals cell 1 half times 2 times sin space open parentheses straight pi over 4 plus straight theta close parentheses times sin space open parentheses straight pi over 4 minus straight theta close parentheses end cell row blank equals cell 1 half times open parentheses 2 times sin space open parentheses straight pi over 4 plus straight theta close parentheses times sin space open parentheses straight pi over 4 minus straight theta close parentheses close parentheses end cell row blank equals cell 1 half open square brackets negative cos space open parentheses open parentheses straight pi over 4 plus straight theta close parentheses plus open parentheses straight pi over 4 minus straight theta close parentheses close parentheses plus cos space open parentheses open parentheses straight pi over 4 plus straight theta close parentheses minus open parentheses straight pi over 4 minus straight theta close parentheses close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative cos space open parentheses straight pi over 4 up diagonal strike plus straight theta end strike plus straight pi over 4 up diagonal strike negative straight theta end strike close parentheses plus cos space open parentheses up diagonal strike straight pi over 4 end strike plus straight theta up diagonal strike negative straight pi over 4 end strike plus straight theta close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative cos space open parentheses fraction numerator 2 straight pi over denominator 4 end fraction close parentheses plus cos space 2 straight theta close square brackets end cell row blank equals cell 1 half open parentheses negative cos space 90 degree plus cos space 2 straight theta close parentheses end cell row blank equals cell 1 half open parentheses 0 plus cos space 2 straight theta close parentheses end cell row blank equals cell 1 half open parentheses cos space 2 straight theta close parentheses end cell row blank equals cell 1 half space cos space 2 straight theta end cell end table end style 

Jadi, terbukti bahwa sin space open parentheses straight pi over 4 plus straight theta close parentheses times sin space open parentheses straight pi over 4 minus straight theta close parentheses equals 1 half space cos space 2 straight theta.

0

Roboguru

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