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Jika cos(x+50∘)=a, dengan 0∘<x<45∘, nilai dari sin(x+20∘) adalah...

Pertanyaan

Jika cos space left parenthesis x plus 50 degree right parenthesis equals a comma dengan 0 degree less than x less than 45 degree, nilai dari sin space left parenthesis x plus 20 degree right parenthesis adalah...

  1. fraction numerator square root of 1 minus a squared end root minus a over denominator 2 end fraction 

  2. fraction numerator square root of 3 open parentheses 1 minus a squared close parentheses end root minus a over denominator 2 end fraction 

  3. fraction numerator square root of 3 open parentheses 1 minus a squared close parentheses end root plus a over denominator 2 end fraction 

  4. fraction numerator square root of 2 open parentheses 1 minus a squared close parentheses end root minus a over denominator 2 end fraction 

  5. fraction numerator square root of 2 open parentheses 1 minus a squared close parentheses end root plus a over denominator 2 end fraction 

Pembahasan Soal:

Diketahui cos space left parenthesis x plus 50 degree right parenthesis equals a, misalkan x plus 20 degree equals y, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis x plus 50 degree right parenthesis end cell equals a row cell cos space left parenthesis x plus 20 degree plus 30 degree right parenthesis end cell equals a row cell cos space left parenthesis y plus 30 degree right parenthesis end cell equals a end table

dengan segitiga siku-siku :

cos space left parenthesis y plus 30 degree right parenthesis equals samping over miring equals a over 1

Maka, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses y plus 30 degree close parentheses end cell equals cell depan over miring end cell row blank equals cell fraction numerator square root of 1 minus a squared end root over denominator 1 end fraction end cell row blank equals cell square root of 1 minus a squared end root end cell end table

Selanjutnya, ingat rumus penjumlahan sudut pada cosinus, dan rumus sudut rangkap:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis A plus B right parenthesis end cell equals cell cos space A space cos space B minus sin space A space sin space B end cell row cell sin space left parenthesis A plus B right parenthesis end cell equals cell sin space A space cos space B plus space cos space A space sin space B end cell end table

Berdasarkan rumus tersebut, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis y plus 30 degree right parenthesis end cell equals a row cell cos space y space cos space 30 degree minus sin space y space sin space 30 degree end cell equals a row cell cos space y times fraction numerator square root of 3 over denominator 2 end fraction minus sin space y times 1 half end cell equals a row cell square root of 3 cos space y minus sin space y end cell equals cell 2 a... left parenthesis 1 right parenthesis end cell row blank blank blank row cell sin space left parenthesis y plus 30 degree right parenthesis end cell equals cell square root of 1 minus a squared end root end cell row cell sin space y space cos space 30 degree plus cos space y space sin space 30 degree end cell equals cell square root of 1 minus a squared end root end cell row cell sin space y times fraction numerator square root of 3 over denominator 2 end fraction plus cos space y times 1 half end cell equals cell square root of 1 minus a squared end root end cell row cell cos space y plus square root of 3 sin space y end cell equals cell 2 square root of 1 minus a squared end root... left parenthesis 2 right parenthesis end cell end table

Selanjutnya eliminasi cos space y pada persamaan (1) dan (2),

begin mathsize 12px style square root of 3 cos space y minus sin space y equals 2 a space space space space space space space space space space space space space open vertical bar cross times 1 close vertical bar space space space square root of 3 cos space y minus sin space y equals 2 a space space cos space y plus square root of 3 sin space y equals 2 square root of 1 minus a squared end root space space open vertical bar cross times square root of 3 close vertical bar space bottom enclose square root of 3 cos space y plus 3 sin space y equals 2 square root of 3 square root of 1 minus a squared end root space subscript minus space end enclose space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 4 space sin space y equals 2 square root of 3 open parentheses 1 minus a squared close parentheses end root minus 2 a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space sin space y equals fraction numerator 2 square root of 3 open parentheses 1 minus a squared close parentheses end root minus 2 a over denominator 4 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space sin space y equals fraction numerator square root of 3 open parentheses 1 minus a squared close parentheses end root minus a over denominator 2 end fraction end style

Jadi, jawaban yang tepat adalah B.

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Janatu

Mahasiswa/Alumni Universitas Riau

Terakhir diupdate 29 Agustus 2021

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Pertanyaan yang serupa

Nilai dari tan240∘sin75∘+cos75∘​=....

Pembahasan Soal:

Ingat rumus jumlah dua sudut pada sinus yaitu

sin open parentheses A plus B close parentheses equals sin space A cos space B plus cos space A sin space B

75 degree equals 45 degree plus 30 degree maka adalah sin space 75 degree equals sin open parentheses 45 plus 30 close parentheses degree sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 75 degree end cell equals cell sin open parentheses 45 plus 30 close parentheses degree end cell row blank equals cell sin space 45 degree cos space 30 degree plus cos space 45 degree sin space 30 end cell row blank equals cell 1 half square root of 2 times 1 half square root of 3 plus 1 half square root of 2 times 1 half end cell row blank equals cell 1 fourth square root of 6 plus 1 fourth square root of 2 end cell end table

Ingat rumus jumlah dua sudut pada cosinus yaitu

cos open parentheses A plus B close parentheses equals cos space A cos space B minus sin space A sin space B

75 degree equals 45 degree plus 30 degree maka adalah cos space 75 degree equals cos open parentheses 45 plus 30 close parentheses degree sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 75 degree end cell equals cell cos open parentheses 45 plus 30 close parentheses degree end cell row blank equals cell cos space 45 degree cos space 30 degree minus sin space 45 degree sin space 30 degree end cell row blank equals cell 1 half square root of 2 times 1 half square root of 3 minus 1 half square root of 2 times 1 half end cell row blank equals cell fraction numerator square root of 6 over denominator 4 end fraction minus fraction numerator square root of 2 over denominator 4 end fraction end cell end table

Sehingga fraction numerator sin space 75 degree plus cos space 75 degree over denominator tan space 240 degree end fraction adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space 75 degree plus cos space 75 degree over denominator tan space 240 degree end fraction end cell equals cell fraction numerator begin display style fraction numerator square root of 6 over denominator 4 end fraction end style plus begin display style fraction numerator square root of 2 over denominator 4 end fraction end style plus begin display style fraction numerator square root of 6 over denominator 4 end fraction end style minus begin display style fraction numerator square root of 2 over denominator 4 end fraction end style over denominator tan open parentheses 180 plus 60 close parentheses degree end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 square root of 6 over denominator 4 end fraction end style over denominator tan space 60 degree end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 square root of 6 over denominator 4 end fraction end style over denominator square root of 3 end fraction end cell row blank equals cell fraction numerator 2 square root of 6 over denominator 4 square root of 3 end fraction end cell row blank equals cell fraction numerator square root of 2 over denominator 2 end fraction end cell end table

Jadi fraction numerator sin space 75 degree plus cos space 75 degree over denominator tan space 240 degree end fraction equals 1 half square root of 2.

Oleh karena itu, jawaban yang benar adalah D.

1

Roboguru

Sederhanakanlah bentuk berikut sin(180∘+α)+cos(90∘+α∘)+sin(270∘+α∘)

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis a plus b right parenthesis end cell equals cell sin space a space cos space b plus cos space a space sin space b end cell row cell cos space left parenthesis a plus b right parenthesis end cell equals cell cos space a space cos space b space minus sin space a space sin space b end cell end table

maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis 180 degree plus alpha degree right parenthesis end cell equals cell sin space 180 degree space cos space alpha degree plus cos space 180 degree space sin space alpha degree end cell row blank equals cell 0 times cos space alpha degree plus left parenthesis negative 1 right parenthesis sin space alpha degree end cell row blank equals cell negative sin space alpha degree end cell row blank blank blank row cell cos space left parenthesis 90 degree plus alpha degree right parenthesis end cell equals cell cos space 90 degree space cos space alpha degree minus sin space 90 degree space sin space alpha degree end cell row blank equals cell 0 times cos space alpha degree minus 1 sin space alpha degree end cell row blank equals cell negative sin space alpha degree end cell row blank blank blank row cell sin space left parenthesis 270 degree plus alpha degree right parenthesis end cell equals cell sin space 270 degree space cos space alpha degree plus cos space 270 degree space sin space alpha degree end cell row blank equals cell negative 1 times cos space alpha degree plus 0 times sin space alpha degree end cell row blank equals cell negative cos space alpha degree end cell end table

lalu substitusikan hasil yang diatas ke persamaan yang berada pada soal

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin open parentheses 180 to the power of ring operator plus alpha degree close parentheses plus cos open parentheses 90 to the power of ring operator plus alpha to the power of ring operator close parentheses plus sin open parentheses 270 to the power of ring operator plus alpha to the power of ring operator close parentheses end cell row blank equals cell negative sin space alpha to the power of ring operator plus left parenthesis negative sin space alpha to the power of ring operator right parenthesis plus left parenthesis negative cos space alpha to the power of ring operator right parenthesis end cell row blank equals cell negative sin space alpha to the power of ring operator minus sin space alpha to the power of ring operator minus cos space alpha to the power of ring operator end cell row blank equals cell negative 2 space sin space alpha to the power of ring operator minus cos space alpha to the power of ring operator end cell end table

jadi, bentuk sederhananya adalah Error converting from MathML to accessible text.

0

Roboguru

Jika A+B+C=π, tunjukkan bahwa: 2sin(A+B)cos(A−B)−2sin(A+B)cos(A+B)=4sin(A)sin(B)sin(C)

Pembahasan Soal:

Diketahui A+B+C=π, akan ditunjukkan 

2sin(A+B)cos(AB)2sin(A+B)cos(A+B)=4sin(A)sin(B)sin(C)

Ingat bahwa 

cos(A+B)=cosAcosBsinAsinBcos(AB)=cosAcosB+sinAsinB 

Diperhatikan

A+B+CA+B==ππC 

Diperoleh perhitungan

=======2sin(A+B)cos(AB)2sin(A+B)cos(A+B)2sin(A+B)(cos(AB)cos(A+B))2sin(A+B)(cosAcosB+sinAsinB(cosAcosBsinAsinB))2sin(A+B)(cosAcosB+sinAsinBcosAcosB+sinAsinB)2sin(A+B)(2sinAsinB)2sin(πC)(2sinAsinB)2sinC(2sinAsinB)4sin(A)sin(B)sin(C) 

Dengan demikian, terbukti bahwa 

2sin(A+B)cos(AB)2sin(A+B)cos(A+B)=4sin(A)sin(B)sin(C)

 

0

Roboguru

Dengan menggunakan formula  cos(α+β)=cos(α)cos(β)−sin(α)sin(β)  tunjukkan kebenaran formula dibawah ini.  sin(α+β)=sin(α)cos(β)+cos(α)sin(β)

Pembahasan Soal:

Diketahui cos(α+β)=cos(α)cos(β)sin(α)sin(β)

Akan ditunjukkan sin(α+β)=sin(α)cos(β)+cos(α)sin(β)

Ingat bahwa sin(α)=cos(90α)

Perhatikan perhitungan berikut

sin(α+β)=====cos(90(α+β))cos(90αβ)cos((90α)β)cos(90α)cos(β)+sin(90α)sin(β)sin(α)cos(β)+cos(α)sin(β)  

Dengan demikian, terbukti sin(α+β)=sin(α)cos(β)+cos(α)sin(β).

0

Roboguru

Buktikan setiap identitas trigonometri berikut. b. sin(A+B)+sin(A−B)cos(A−B)−cos(A+B)​=tanB

Pembahasan Soal:

Ingat rumus jumlah dan selisih dua sudut pada cosinus yaitu

Error converting from MathML to accessible text.

Maka cos open parentheses text A end text minus text B end text close parentheses minus cos open parentheses text A end text plus text B end text close parentheses yaitu

table row cell cos open parentheses A minus B close parentheses end cell cell equals cos space A cos space B plus sin space A sin space B end cell row cell cos open parentheses A plus B close parentheses end cell cell equals cos space A cos space B minus sin space A sin space B space space space minus end cell row cell cos open parentheses A minus B close parentheses minus cos open parentheses A plus B close parentheses end cell cell equals 2 space sin space A sin space B end cell end table

Ingat juga rumus jumlah dan selisih dua sudut pada sinus yaitu

Error converting from MathML to accessible text.

Maka sin open parentheses text A end text plus text B end text close parentheses plus sin open parentheses text A end text minus text B end text close parentheses sebagai berikut.

table row cell sin open parentheses A plus B close parentheses end cell cell equals sin space A cos space B plus cos space A sin space B end cell row cell sin open parentheses A minus B close parentheses end cell cell equals sin space A cos space B minus cos space A sin space B space space space space plus end cell row cell sin open parentheses A plus B close parentheses plus sin open parentheses A minus B close parentheses end cell cell equals 2 space sin space A cos space B end cell end table

Sehingga fraction numerator cos open parentheses text A end text minus text B end text close parentheses minus cos open parentheses text A end text plus text B end text close parentheses over denominator sin open parentheses text A end text plus text B end text close parentheses plus sin open parentheses text A end text minus text B end text close parentheses end fraction equals tan space text B end text diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos open parentheses text A end text minus text B end text close parentheses minus cos open parentheses text A end text plus text B end text close parentheses over denominator sin open parentheses text A end text plus text B end text close parentheses plus sin open parentheses text A end text minus text B end text close parentheses end fraction end cell equals cell fraction numerator up diagonal strike 2 sin space A end strike sin B over denominator up diagonal strike 2 sin space A end strike cos space B end fraction end cell row blank equals cell fraction numerator sin space B over denominator cos space B end fraction end cell row blank equals cell tan space B end cell end table

Jadi berdasarkan hitungan di atas terbukti bahwa fraction numerator cos open parentheses text A end text minus text B end text close parentheses minus cos open parentheses text A end text plus text B end text close parentheses over denominator sin open parentheses text A end text plus text B end text close parentheses plus sin open parentheses text A end text minus text B end text close parentheses end fraction equals tan space text B end text.

 

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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