Roboguru

Nilai dari cos55∘+cos25∘tan75∘(sin55∘−sin25∘)​=...

Pertanyaan

Nilai dari fraction numerator tan space 75 degree open parentheses sin space 55 degree minus sin space 25 degree close parentheses over denominator cos space 55 degree plus cos space 25 degree end fraction equals... 

  1. negative 1

  2. negative 1 half

  3. 0

  4. 1 half

  5. 1

Pembahasan Soal:

Ingat kembali rumus-rumus pengurangan sinus dan penjumlahan cosinus berikut.

  1. sin space alpha minus sin space beta equals 2 space cos space open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses space sin space open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses
  2. cos space alpha plus cos space beta equals 2 space cos space open parentheses fraction numerator alpha plus beta over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses

Begitu juga dengan rumus perkalian sinus dan kosinus:

  1. 2 space sin space alpha space sin space beta equals cos space open parentheses alpha minus beta close parentheses minus cos space open parentheses alpha plus beta close parentheses
  2. 2 space cos space alpha space cos space beta equals cos space open parentheses alpha plus beta close parentheses plus cos space open parentheses alpha minus beta close parentheses

Berdasarkan rumus-rumus di atas, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 75 degree open parentheses sin space 55 degree minus sin space 25 degree close parentheses over denominator cos space 55 degree plus cos space 25 degree end fraction end cell equals cell tan space 75 degree fraction numerator up diagonal strike 2 space cos space open parentheses begin display style fraction numerator 55 degree plus 25 degree over denominator 2 end fraction end style close parentheses end strike space sin space open parentheses begin display style fraction numerator 55 degree minus 25 degree over denominator 2 end fraction end style close parentheses over denominator up diagonal strike 2 space cos space open parentheses begin display style fraction numerator 55 degree plus 25 degree over denominator 2 end fraction end style close parentheses end strike space cos space open parentheses begin display style fraction numerator 55 degree minus 25 degree over denominator 2 end fraction end style close parentheses end fraction end cell row blank equals cell tan space 75 degree fraction numerator sin space 15 degree over denominator cos space 15 degree end fraction end cell row blank equals cell fraction numerator sin space 75 degree space sin space 15 degree over denominator cos space 75 degree space cos space 15 degree end fraction end cell row blank equals cell fraction numerator up diagonal strike begin display style 1 half end style end strike open parentheses cos space 60 degree minus cos space 90 degree close parentheses over denominator up diagonal strike begin display style 1 half end style end strike open parentheses cos space 90 degree plus cos space 60 degree close parentheses end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style minus 0 over denominator 0 plus begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style bevelled 1 half end style over denominator begin display style bevelled 1 half end style end fraction end cell row blank equals 1 end table 

Dengan demikian, hasil dari fraction numerator tan space 75 degree open parentheses sin space 55 degree minus sin space 25 degree close parentheses over denominator cos space 55 degree plus cos space 25 degree end fraction adalah 1.

Oleh karena itu, jawaban yang benar adalah E.

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

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

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

Hitunglah: tan69∘+tan33∘−tan39∘−tan3∘.

Pembahasan Soal:

Ingat bahwa:

  • tan space A equals fraction numerator sin space A over denominator cos space B end fraction 
  • sin left parenthesis A minus B right parenthesis equals sin space A space cos space B minus cos space A space sin space B 
  • 2 space cos space A space cos space B equals cos space left parenthesis A plus B right parenthesis plus cos space left parenthesis A minus B right parenthesis 
  • cos space left parenthesis 90 degree plus A right parenthesis equals negative sin space A 
  • sin space 18 degree equals fraction numerator square root of 5 minus 1 over denominator 4 end fraction 
  • cos space 36 degree equals fraction numerator square root of 5 plus 1 over denominator 4 end fraction 

Mencari nilai tan space 69 degree plus tan space 33 degree minus tan space 39 degree minus tan space 3 degree:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell tan open parentheses 69 degree close parentheses plus tan open parentheses 33 degree close parentheses minus tan open parentheses 39 degree close parentheses minus tan open parentheses 3 degree close parentheses end cell row blank equals cell tan open parentheses 69 degree close parentheses minus tan open parentheses 39 degree close parentheses plus tan open parentheses 33 degree close parentheses minus tan open parentheses 3 degree close parentheses end cell row blank equals cell open parentheses fraction numerator sin space 69 degree over denominator cos space 69 degree end fraction minus fraction numerator sin space 39 degree over denominator cos space 39 degree end fraction close parentheses plus open parentheses fraction numerator sin space 33 degree over denominator cos space 33 degree end fraction minus fraction numerator sin space 3 degree over denominator cos space 3 degree end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator sin space 69 degree cos space 39 degree minus sin space 69 degree cos space 39 degree over denominator cos space 69 degree cos space 39 degree end fraction close parentheses end cell row blank blank cell plus open parentheses fraction numerator sin space 33 degree cos space 3 degree minus sin space 33 degree cos space 3 degree over denominator cos space 33 degree cos space 3 degree end fraction close parentheses end cell row blank equals cell fraction numerator sin open parentheses 69 degree minus 39 degree close parentheses over denominator cos space 69 degree cos space 39 degree end fraction plus fraction numerator sin open parentheses 33 degree minus 3 degree close parentheses over denominator cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator sin open parentheses 30 degree close parentheses over denominator cos space 69 degree cos space 39 degree end fraction plus fraction numerator sin open parentheses 30 degree close parentheses over denominator cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 cos space 69 degree cos space 39 degree end fraction plus fraction numerator 1 over denominator 2 cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space open parentheses 69 degree plus 39 degree close parentheses plus cos space open parentheses 69 degree minus 39 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 33 degree plus 3 degree close parentheses plus cos space open parentheses 33 degree minus 3 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space open parentheses 108 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 36 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator negative sin space open parentheses 18 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 30 degree close parentheses plus cos space open parentheses 36 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator negative begin display style fraction numerator square root of 5 minus 1 over denominator 4 end fraction end style plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction plus fraction numerator 1 over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction plus fraction numerator square root of 5 plus 1 over denominator 4 end fraction end style end fraction end cell row blank equals cell fraction numerator 4 over denominator negative square root of 5 plus 1 plus 2 square root of 3 end fraction plus fraction numerator 4 over denominator 2 square root of 3 plus square root of 5 plus 1 end fraction end cell row blank equals cell 4 open parentheses fraction numerator 1 over denominator negative square root of 5 plus 1 plus 2 square root of 3 end fraction plus fraction numerator 1 over denominator 2 square root of 3 plus square root of 5 plus 1 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator open parentheses 2 square root of 3 plus 1 close parentheses plus square root of 5 minus square root of 5 plus open parentheses 1 plus 2 square root of 3 close parentheses over denominator open square brackets negative square root of 5 plus open parentheses 1 plus 2 square root of 3 close parentheses close square brackets open square brackets open parentheses 2 square root of 3 plus 1 close parentheses plus square root of 5 close square brackets end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 2 open parentheses 2 square root of 3 plus 1 close parentheses over denominator open parentheses negative square root of 5 close parentheses squared plus open parentheses 1 plus 2 square root of 3 close parentheses squared end fraction close parentheses end cell row blank equals cell fraction numerator 8 open parentheses 2 square root of 3 plus 1 close parentheses over denominator 12 plus 1 plus 4 square root of 3 minus 5 end fraction end cell row blank equals cell 2 open parentheses 3 square root of 3 minus 4 close parentheses end cell row blank almost equal to cell 2 comma 392 end cell end table  

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

Nilai eksak dari (cos20∘cos40∘cos80∘) adalah . . . .

Pembahasan Soal:

Ingat kembali bahwa:

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

Sehingga diperoleh perhitungan:

cos20cos40cos80=(cos20cos40)cos80=21[2cos20cos40]cos80=21[[cos(40+20)+cos(4020]cos80=21[cos60+cos20]cos80=21[21+cos20]cos80=21cos80[21+cos20]=41cos80+21cos80cos20=41cos80+2121(2cos80cos20)=41cos80+2121[cos(80+20)+cos(8020)]=41cos80+2121[cos100+cos60]=41cos80+41[cos100+cos60]=41cos80+41cos100+41cos60=41cos80+41cos100+4121=41(cos80+cos100)+81=41(cos(280+100)cos210080)+81=41(cos90cos10)+81=41(0cos10)+81=81    

Dengan demikian, nilai dari (cos20cos40cos80) adalah 81

Jadi jawaban yang tepat adalah E.

0

Roboguru

Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. g. 5cos2x+3cosx−2=0

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.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved