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Tentukan nilai Rdanα dari setiap persamaan di bawah ini. 8sin(θ)+6cos(θ)=Rsin(θ+α)

Pertanyaan

Tentukan nilai Rdanα dari setiap persamaan di bawah ini.

8sin(θ)+6cos(θ)=Rsin(θ+α) 

Pembahasan Soal:

Diketahui 8sin(θ)+6cos(θ)=Rsin(θ+α), akan dicari Rdanα

Ingat bahwa

sin(θ+α)=sinθcosα+cosθsinαAsinθ+Bcosθ=Rsin(θ+α),denganR=A2+B2 

Diperhatikan

8sin(θ)+6cos(θ)=Rsin(θ+α)8sin(θ)+6cos(θ)=Rsinθcosα+Rcosθsinα 

Diperoleh

8=Rcosα6=Rsinα 

Lebih lanjut, diperoleh

6 equals R space sin space alpha 8 equals R space cos space alpha horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike horizontal strike blank end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike end strike space divided by 2 over 3 equals fraction numerator R space sin space alpha over denominator R space cos space alpha end fraction 2 over 3 equals tan space alpha alpha almost equal to 33 comma 69 degree 

dan

66RR====RsinαRsin33,69sin33,69610,81 

Dengan demikian, diperoleh R=10,81danα33,69 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Firmansyah

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 30 Agustus 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Buktikan bahwa: b.

Pembahasan Soal:

Ingat rumus jumlah dua sudut pada sinus dan cosinus yaitu

sin open parentheses text A end text plus text B end text close parentheses equals sin space text A end text times cos space text B end text plus cos space text A end text space sin space text B end text
cos open parentheses A plus B close parentheses equals cos space A cos space B minus sin space A sin space B

Sehingga sin space 2 x equals sin open parentheses x plus x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 x end cell equals cell sin open parentheses x plus x close parentheses end cell row blank equals cell sin space x space cos space x plus cos space x space sin space x end cell row blank equals cell 2 space sin space x space cos space x end cell end table

Dan cos space 2 x equals cos open parentheses x plus x close parentheses yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 x end cell equals cell cos open parentheses x plus x close parentheses end cell row blank equals cell cos space x cos space x minus sin space x sin space x end cell row blank equals cell cos squared x minus sin squared x end cell end table

Maka sin space 3 x equals sin open parentheses 2 x plus x close parentheses yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 3 x end cell equals cell sin open parentheses 2 x plus x close parentheses end cell row blank equals cell sin space 2 x space cos space x plus cos space 2 x space sin space x end cell row blank equals cell open parentheses 2 space sin space x space cos space x close parentheses cos space x plus open parentheses cos squared x minus sin squared x close parentheses sin space x end cell row blank equals cell 2 space sin space x space cos squared space x plus cos squared x sin space x minus sin cubed x end cell row blank equals cell 3 space sin space x space cos squared space x minus sin cubed x end cell row blank equals cell 3 space sin space x space open parentheses 1 minus sin squared x close parentheses minus sin cubed x end cell row blank equals cell 3 space sin space x minus 3 space sin cubed x minus sin cubed x end cell row blank equals cell 3 space sin space x minus 4 space sin cubed x end cell end table


Jadi berdasarkan hitungan di atas terbukti bahwa sin space 3 x equals 3 space sin space x minus 4 space sin cubed x.

0

Roboguru

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

Pembahasan Soal:

Diketahui cos(x+50)=a, misalkan x+20=y, maka

cos(x+50)cos(x+20+30)cos(y+30)===aaa

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:

cos(A+B)sin(A+B)==cosAcosBsinAsinBsinAcosB+cosAsinB

Berdasarkan rumus tersebut, maka

cos(y+30)cosycos30sinysin30cosy23siny213cosysinysin(y+30)sinycos30+cosysin30siny23+cosy21cosy+3siny========aaa2a...(1)1a21a21a221a2...(2)

Selanjutnya eliminasi cosy pada persamaan (1) dan (2),

3cosysiny=2a×13cosysiny=2acosy+3siny=21a2×33cosy+3siny=231a24siny=23(1a2)2asiny=423(1a2)2asiny=23(1a2)a

Jadi, jawaban yang tepat adalah B.

 

0

Roboguru

Nilai dari  adalah ...

Pembahasan Soal:

Ingat kembali beberapa konsep pada trigonometri berikut:

Rumus trigonometri jumlah sudut 

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

Sudut berelasi pada trigonometri sin

sin space open parentheses 180 degree minus x degree close parentheses equals sin space x degree

Dari soal, didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 75 degree space cos space 45 degree plus cos space 75 degree space sin space 45 degree end cell equals cell sin space open parentheses 75 degree plus 45 degree close parentheses end cell row blank equals cell sin space 120 degree end cell row blank equals cell sin space open parentheses 180 degree minus 60 degree close parentheses end cell row blank equals cell sin space 60 degree end cell row blank equals cell 1 half square root of 3 end cell end table

Jadi, nilai dari sin space 75 degree space cos space 45 degree plus cos space 75 degree space sin space 45 degree adalah 1 half square root of 3.

0

Roboguru

Diketahui  dan , dengan sudut A dan sudut B lancip. Hitunglah nilai: a. ;

Pembahasan Soal:

A dan B sudut lancip maka terletak di kuadran I, dengan demikian A dan B akan bernilai positif.

Menentukan nilai cos space A, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space A end cell equals cell 1 half rightwards double arrow depan over miring end cell row samping equals cell square root of miring squared minus depan squared end root end cell row blank equals cell square root of 2 squared minus 1 squared end root end cell row blank equals cell square root of 4 minus 1 end root end cell row blank equals cell square root of 3 end cell end table

Sehingga: cos space A equals samping over miring rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction.

Menentukan nilai cos space B, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space B end cell equals cell 1 third rightwards double arrow depan over miring end cell row samping equals cell square root of miring squared minus depan squared end root end cell row blank equals cell square root of 3 squared minus 1 squared end root end cell row blank equals cell square root of 9 minus 1 end root end cell row blank equals cell square root of 8 end cell end table

Sehingga: cos space B equals samping over miring rightwards double arrow fraction numerator square root of 8 over denominator 3 end fraction.

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses A plus B close parentheses end cell equals cell sin space A times cos space B plus sin space B times cos space A end cell row blank equals cell 1 half times fraction numerator square root of 8 over denominator 3 end fraction plus 1 third times fraction numerator square root of 3 over denominator 2 end fraction end cell row blank equals cell fraction numerator square root of 8 over denominator 6 end fraction plus fraction numerator square root of 3 over denominator 6 end fraction end cell row blank equals cell 1 over 6 open parentheses square root of 8 plus square root of 3 close parentheses end cell end table

Maka, nilai sin space open parentheses A plus B close parentheses adalah 1 over 6 open parentheses square root of 8 plus square root of 3 close parentheses.

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Roboguru

....

Pembahasan Soal:

Ingat bahwa

begin mathsize 14px style 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis A plus B right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A 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 blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A 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 blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos invisible function application left parenthesis A plus B right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A 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 blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A 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 blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table end style   

Sehingga

begin mathsize 14px style 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis A plus B plus C right parenthesis end cell end table equals 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis open parentheses A plus B close parentheses plus C right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table equals 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis A plus B right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis A plus B right parenthesis times end cell end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style 

Kita sederhanakan dulu bentuk begin mathsize 14px style sin open parentheses A plus B close parentheses times cos space C end style.

begin mathsize 14px style sin space left parenthesis A plus B right parenthesis times cos C equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis sin invisible function application A space cos invisible function application B plus cos invisible function application A space sin invisible function application B right parenthesis times cos C end cell end table equals 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell space cos end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell B space end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell C space end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell A space end cell end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell B space end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell C space end cell end table end style     

Kemudian kita sederhanakan pula bentuk begin mathsize 14px style cos open parentheses A plus B close parentheses times sin C end style.

begin mathsize 14px style cos open parentheses A plus B close parentheses times sin space C equals open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell space cos end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell A space end cell end table 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 invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table close parentheses times 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 invisible function application end table C equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos invisible function application A space cos invisible function application B space sin invisible function application C minus sin invisible function application A space sin invisible function application B space sin invisible function application C end cell end table end style 

Sehingga diperoleh,

begin mathsize 14px style sin space open parentheses A plus B plus C close parentheses equals sin space A space cos space B space cos space C plus cos space A space sin space B space cos space C plus 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 cos space A space cos space B space sin space C minus sin space A space sin space B space sin space C space space end style 

Jadi, jawaban yang tepat adalah B.  

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Roboguru

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