Roboguru

(sin75∘cos15∘)−(cos75∘sin15∘)=...

Pertanyaan

open parentheses sin space 75 degree space cos space 15 degree close parentheses minus open parentheses cos space 75 degree space sin space 15 degree close parentheses equals...

  1. 1 half square root of 2

  2. square root of 2

  3. 1 plus square root of 2

  4. 1 half square root of 3

  5. negative 1 half square root of 3

Pembahasan Soal:

Rumus perkalian sinus dan kosinus adalah sebagai berikut.

open parentheses straight i close parentheses space space 2 space sin space A space cos space B equals sin space open parentheses A plus B close parentheses plus sin space open parentheses A minus B close parentheses left parenthesis ii right parenthesis space space 2 space cos space A space sin space B equals sin space open parentheses A plus B close parentheses minus sin space open parentheses A minus B close parentheses left parenthesis iii right parenthesis space 2 space cos space A space cos space B equals cos space open parentheses A plus B close parentheses plus cos space open parentheses A minus B close parentheses left parenthesis iv right parenthesis space 2 space sin space A space sin space B equals cos space open parentheses A minus B close parentheses minus cos space open parentheses A plus B close parentheses

Berdasarkan rumus no (i) dan (ii) di atas, maka:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin space 75 degree space cos space 15 degree close parentheses minus open parentheses cos space 75 degree space sin space 15 degree close parentheses end cell row blank equals cell 1 half open parentheses sin space open parentheses 75 degree plus 15 degree close parentheses plus sin space open parentheses 75 degree minus 15 degree close parentheses close parentheses end cell row blank blank cell negative 1 half space open parentheses sin space open parentheses 75 degree plus 15 degree close parentheses minus sin space open parentheses 75 degree minus 15 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses space sin space 90 degree plus sin space 60 degree close parentheses minus 1 half open parentheses sin space 90 degree minus sin space 60 degree close parentheses end cell row blank equals cell 1 half open parentheses 1 plus 1 half square root of 3 close parentheses minus 1 half open parentheses 1 minus 1 half square root of 3 close parentheses end cell row blank equals cell 1 half plus 1 fourth square root of 3 minus 1 half plus 1 fourth square root of 3 end cell row blank equals cell 2 over 4 square root of 3 end cell row blank equals cell 1 half square root of 3 end cell end table

Dengan demikian, hasil dari open parentheses sin space 75 degree space cos space 15 degree close parentheses minus open parentheses cos space 75 degree space sin space 15 degree close parentheses adalah 1 half square root of 3.

Oleh karena itu, jawaban yang benar adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Mahasiswa/Alumni Universitas Padjadjaran

Terakhir diupdate 07 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Nilai dari 2sin75∘⋅cos15∘ adalah ,,,

Pembahasan Soal:

Rumus perkalian sinus dan kosinus adalah sebagai berikut.

open parentheses straight i close parentheses space space 2 space sin space A space cos space B equals sin space open parentheses A plus B close parentheses plus sin space open parentheses A minus B close parentheses left parenthesis ii right parenthesis space space 2 space cos space A space sin space B equals sin space open parentheses A plus B close parentheses minus sin space open parentheses A minus B close parentheses left parenthesis iii right parenthesis space 2 space cos space A space cos space B equals cos space open parentheses A plus B close parentheses plus cos space open parentheses A minus B close parentheses left parenthesis iv right parenthesis space 2 space sin space A space sin space B equals cos space open parentheses A minus B close parentheses minus cos space open parentheses A plus B close parentheses

Berdasarkan rumus no (i) di atas, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space sin space 75 degree times cos space 15 degree end cell equals cell sin space open parentheses 75 degree plus 15 degree close parentheses plus sin space open parentheses 75 degree minus 15 degree close parentheses end cell row blank equals cell sin space 90 degree plus sin space 60 degree end cell row blank equals cell 1 plus 1 half square root of 3 end cell end table

Dengan demikian, hasil dari 2 space sin space 75 degree times cos space 15 degree adalah 1 plus 1 half square root of 3.

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Tentukan nilai dari: sin35∘cos5∘−cos35∘cos85∘.

Pembahasan Soal:

Gunakan konsep rumus perkalian trigonometri dan sudut relasi open parentheses 90 degree minus x close parentheses.

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 space sin space straight A space cos space straight B end cell equals cell sin space open parentheses straight A plus straight B close parentheses plus sin space open parentheses straight A minus straight B close parentheses end cell row atau blank blank row cell sin space straight A space cos space straight B end cell equals cell 1 half open parentheses sin space open parentheses straight A plus straight B close parentheses plus sin space open parentheses straight A minus straight B close parentheses close parentheses end cell row blank blank blank row cell 2 space cos space straight A space cos space straight B end cell equals cell cos space open parentheses straight A plus straight B close parentheses plus cos space open parentheses straight A minus straight B close parentheses end cell row atau blank blank row cell cos space straight A space cos space straight B end cell equals cell 1 half open parentheses cos space open parentheses straight A plus straight B close parentheses plus cos space open parentheses straight A minus straight B close parentheses close parentheses end cell row blank blank blank row cell sin space open parentheses 90 degree minus alpha close parentheses end cell equals cell cos space alpha end cell row cell cos space open parentheses negative alpha close parentheses end cell equals cell cos space alpha end cell end table

Ingat kembali nilai trigonometri sudut istimewa 30 degree dan sudut istimewa di kuadran II 120 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space 30 degree end cell equals cell 1 half end cell row cell cos space 120 degree end cell equals cell negative 1 half end cell end table

Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell sin space 35 degree space cos space 5 degree minus cos space 35 degree space cos space 85 degree end cell row blank equals cell 1 half open parentheses sin space open parentheses 35 degree plus 5 degree close parentheses plus sin space open parentheses 35 degree minus 5 degree close parentheses close parentheses minus end cell row blank blank cell 1 half open parentheses cos space open parentheses 35 degree plus 85 degree close parentheses plus cos space open parentheses 35 degree minus 85 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses sin space 40 degree plus sin space 30 degree close parentheses minus 1 half open parentheses cos space 120 degree plus cos space open parentheses negative 50 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses sin space open parentheses 90 degree minus 50 degree close parentheses plus 1 half close parentheses minus 1 half open parentheses negative 1 half plus cos space 50 degree close parentheses end cell row blank equals cell 1 half open parentheses cos space 50 degree plus 1 half close parentheses minus 1 half open parentheses negative 1 half plus cos space 50 degree close parentheses end cell row blank equals cell fraction numerator cos space 50 degree over denominator 2 end fraction plus 1 fourth plus 1 fourth minus fraction numerator cos space 50 degree over denominator 2 end fraction end cell row blank equals cell open parentheses 1 fourth plus 1 fourth close parentheses plus open parentheses fraction numerator cos space 50 degree over denominator 2 end fraction minus fraction numerator cos space 50 degree over denominator 2 end fraction close parentheses end cell row blank equals cell 2 over 4 plus 0 end cell row blank equals cell 1 half end cell end table 

Jadi, diperoleh nilai sin space 35 degree space cos space 5 degree minus cos space 35 degree space cos space 85 degree equals 1 half.

0

Roboguru

Nilai cos50∘cos70∘cos20∘​=....

Pembahasan Soal:

0

Roboguru

Nilai dari (tan10∘+tan70∘−tan50∘) adalah ....

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

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

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

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

sinAsinB=2cos21(A+B)sin21(AB)

cosA+cosB=2cos21(A+B)cos21(AB)

Dengan menggunakan konsep di atas, diperoleh hasil:

tan10+tan70tan50=(cos10sin10+cos70sin70)cos50sin50=(cos70cos10cos70sin10+sin70cos10)cos50sin50=22(cos70cos10cos70sin10+sin70cos10)cos50sin50=(2cos70cos102cos70sin10+2sin70cos10)cos50sin50=(cos(70+10)+cos(7010)(sin(70+10)sin(7010))+(sin(70+10)+sin(7010)))cos50sin50=(cos80+cos60(sin80sin60)+(sin80+sin60))cos50sin50=cos80+212sin80cos50sin50=cos80cos50+21cos502sin80cos50sin50(cos80+21)(samakanpenyebut)=cos80cos50+21cos502sin80cos50cos80sin5021sin50=cos80cos50+21cos502sin80cos50cos80sin5021sin5022=2cos80cos50+cos5022sin80cos502cos80sin50sin50=cos(80+50)+cos(8050)+cos502(sin(80+50)+sin(8050))(sin(80+50)sin(8050))sin50=cos130+cos30+cos502(sin130+sin30)(sin130sin30)sin50=cos130+213+cos502(sin130+21)(sin13021)sin50=213+cos130+cos502sin130+1sin130+21sin50=213+cos130+cos5023+sin130sin50=213+(2cos21(130+50)cos21(13050))23+(2cos21(130+50)sin21(13050))=213+(2cos21(180)cos21(80))23+(2cos21(180)sin21(80))=213+(2cos90cos40)23+(2cos90sin40)=213+(20cos40)23+(20sin40)=213+023+0=21323=23÷23=23×32=33=33×33=333=3

Jadi, tan10+tan70tan50=3.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Tanpa menggunakan tabel trigonometri, buktikan: sin52∘sin68∘−sin47∘cos77∘−cos65∘cos81∘=21​

Pembahasan Soal:

Ingat kembali:

2 space sin space straight alpha space cos space straight beta equals sin space open parentheses straight alpha plus straight beta close parentheses plus sin space open parentheses straight alpha minus straight beta close parentheses 2 space cos space straight alpha space cos space straight beta equals cos space open parentheses straight alpha plus straight beta close parentheses minus cos space open parentheses straight alpha minus straight beta close parentheses 2 space sin space straight alpha space sin space straight beta equals negative cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses sin space open parentheses 180 degree minus straight A close parentheses equals sin space straight A cos space open parentheses 90 degree plus straight A close parentheses equals negative sin space straight A 

Akan dibuktikan sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half. 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 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times 2 times sin space 52 degree space sin space 68 degree minus 1 half times 2 times sin space 47 degree space cos space 77 degree minus 1 half times 2 times cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times open parentheses 2 times sin space 52 degree space sin space 68 degree close parentheses minus 1 half times open parentheses 2 times sin space 47 degree space cos space 77 degree close parentheses minus 1 half times open parentheses 2 times cos space 65 degree space cos space 81 degree close parentheses end cell row blank equals cell 1 half open square brackets negative cos space open parentheses 52 degree plus 68 degree close parentheses plus cos space open parentheses 52 degree minus 68 degree close parentheses close square brackets minus 1 half open square brackets sin space open parentheses 47 degree plus 77 degree close parentheses plus sin space open parentheses 47 degree minus 77 degree close parentheses close square brackets minus 1 half open square brackets cos space open parentheses 65 degree plus 81 degree close parentheses plus cos space open parentheses 65 degree minus 81 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative cos space 120 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets minus 1 half open square brackets sin space 124 degree plus sin space open parentheses negative 30 degree close parentheses close square brackets minus 1 half open square brackets cos space 146 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative open parentheses negative 1 half close parentheses minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus sin space 30 degree close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open square brackets 1 half minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus 1 half close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open parentheses 1 half up diagonal strike negative cos space 16 degree end strike minus sin space 124 degree plus 1 half minus cos space 146 degree up diagonal strike plus cos space 16 end strike degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus sin space 124 degree minus cos space 146 degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space 124 degree plus cos space 146 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space open parentheses 180 degree minus 56 degree close parentheses plus cos space open parentheses 90 degree plus 56 degree close parentheses close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses up diagonal strike sin space 56 degree end strike up diagonal strike negative sin space 56 degree end strike close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus 0 close parentheses end cell row blank equals cell 1 half open parentheses 1 close parentheses end cell row blank equals cell 1 half end cell end table end style

Jadi, terbukti bahwa sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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