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Dalam segitiga ABC diketahui sudut A,B,C berhadapan dengan sisi a,b,c. Jika b>c maka b+cb−c​=...

Pertanyaan

Dalam segitiga ABC diketahui sudut A,B,C berhadapan dengan sisi a,b,c. Jika b>c maka b+cbc=... 

  1. cos21Asin21(BC) 

  2. sin21Acos21(BC) 

  3. sin21Atan21(BC) 

  4. tan21Atan21(BC) 

  5. cot21Atan21(BC) 

Pembahasan Soal:

Pada segitiga ABC berlaku aturan sinus yang nilai perbandingannya adalah dua kali panjang jari-jari lingkaran luar segitiga, yaitu : 

sinAa=sinBb=sinCc=2R 

Dari aturan sinus tersebut diperoleh kesamaan : 

sinBb=2Rb=2RsinBsinCc=2Rc=2RsinC 

Substitusi persamaan tersebut pada b+cbc, sehingga diperoleh 

b+cbc=========2RsinB+2RsinC2RsinB2RsinC2R(sinB+sinC)2R(sinBsinC)sinB+sinCsinBsinC2sin21(B+C)cos21(BC)2cos21(B+C)sin21(BC)cot21(B+C)tan21(BC)cot21(180A)tan21(BC)cot(9021A)tan21(BC)tan21Atan21(BC)cot21Atan21(BC) 

Oleh karena itu, jawaban yang benar adalah E.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

L. Rante

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 13 September 2021

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Pertanyaan yang serupa

Jika sin(22a+π​)=53​ maka nilai dari sin(a−π)+cos(−a)=....

Pembahasan Soal:

Ingat rumus jumlah dan selisih sudut trigonometri di bawah ini.

sin(A+B)=sinAcosB+cosAsinB

sin(AB)=sinAcosBcosAsinB

Diketahui sin(22a+π)=53, maka

sin(22a+π)sin(22a+2π)sin(a+2π)sinacos2π+cosasin2πsina0+cosa10+cosacosa=======53535353535353

Perhatikan segitiga siku-siku di bawah ini.

Untuk mencari nilai sina, kita perlu mencari nilai x dengan menggunakan rumus Pythagoras berikut ini.

x2+32x2+9x2x2x=====52252591616=4

Dengan demikian, nilai sina=5x=54.

Selanjutnya kita akan mencari nilai sin(aπ)+cos(a)=51 yaitu

sin(aπ)+cos(a)=sinacosπcosasinπ+cos(a)=sinacosπcosasinπ+cosa=54(1)540+53=54+53=51

Jadi, jika sin(22a+π)=53 maka nilai dari sin(aπ)+cos(a)=51.

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Diketahui system persamaan :  {sin(x+y)=1+51​cosysin(x−y)=−1+cosy​  Dengan 0<y<2π​, maka cos2x=...

Pembahasan Soal:

Ingat! 

sin(A+B)sin(AB)cos2A===sinAcosB+sinBcosAsinAcosBsinBcosA12sin2A 

Selanjutnya 

sin(x+y)sin(x+y)==sinxcosy+cosxsinysinxcosycosxsiny2sinxcosysinx====1+51cosy1+51cosy56cosy53+ 

Jadi, 

cos2x=12sin2x=12(53)2=12518=257

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Diketahui sebuah segitiga  dengan sudut  adalah  dan sudut  adalah . Jika panjang  adalah , maka panjang  adalah ....

Pembahasan Soal:

Perhatikan gambar berikut ini!

 space space space  

Dari gambar tersebut, didapat

table attributes columnalign right center left columnspacing 0px end attributes row cell sin 15 end cell equals cell sin open parentheses 45 minus 30 close parentheses end cell row blank equals cell sin 45 space cos 30 minus cos 45 space sin 30 end cell row blank equals cell open parentheses 1 half square root of 2 close parentheses open parentheses 1 half square root of 3 close parentheses minus open parentheses 1 half square root of 2 close parentheses open parentheses 1 half close parentheses end cell row blank equals cell 1 fourth square root of 2 open parentheses square root of 3 minus 1 close parentheses end cell row cell sin 105 end cell equals cell sin open parentheses 45 plus 60 close parentheses end cell row blank equals cell sin 45 space cos 60 plus cos 45 space sin 60 end cell row blank equals cell open parentheses 1 half square root of 2 close parentheses open parentheses 1 half close parentheses plus open parentheses 1 half square root of 2 close parentheses open parentheses 1 half square root of 3 close parentheses end cell row blank equals cell 1 fourth square root of 2 open parentheses 1 plus square root of 3 close parentheses end cell end table  

Dengan menggunakan aturan sinus,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator BC over denominator sin 15 end fraction end cell equals cell fraction numerator AC over denominator sin 105 end fraction end cell row BC equals cell fraction numerator AC over denominator sin 105 end fraction times sin 15 end cell row BC equals cell 5 times fraction numerator 1 fourth square root of 2 left parenthesis square root of 3 minus 1 right parenthesis over denominator 1 fourth square root of 2 left parenthesis square root of 3 plus 1 right parenthesis end fraction end cell row BC equals cell 5 times fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction space times open parentheses fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction close parentheses end cell row BC equals cell 5 times fraction numerator left parenthesis square root of 3 minus 1 right parenthesis left parenthesis square root of 3 minus 1 right parenthesis over denominator left parenthesis square root of 3 plus 1 right parenthesis left parenthesis square root of 3 minus 1 right parenthesis end fraction end cell row BC equals cell fraction numerator 5 left parenthesis 4 minus 2 square root of 3 right parenthesis over denominator 2 end fraction end cell row BC equals cell fraction numerator 5 times 2 left parenthesis 2 minus square root of 3 right parenthesis over denominator 2 end fraction end cell row BC equals cell 5 left parenthesis 2 minus square root of 3 right parenthesis end cell row blank blank blank end table end style  

Maka panjang begin mathsize 14px style BC end style adalah undefined.

Jadi, jawabannya adalah E.undefined  

0

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

Jika dengan , maka nilai dari  adalah ....

Pembahasan Soal:

Perhatikan ruas kiri terlebih dahulu!

begin mathsize 14px style 2 sin invisible function application x open parentheses sin invisible function application x minus square root of 3 cos invisible function application x close parentheses minus 1 equals 2 sin squared invisible function application x minus 2 square root of 3 sin invisible function application x cos invisible function application x minus 1 equals 2 sin squared invisible function application x minus 1 minus 2 square root of 3 sin invisible function application x cos invisible function application x equals negative open parentheses negative 2 sin squared invisible function application x plus 1 close parentheses minus 2 square root of 3 sin invisible function application x cos invisible function application x equals negative open parentheses 1 minus 2 sin squared invisible function application x close parentheses minus square root of 3 open parentheses 2 sin invisible function application x cos invisible function application x close parentheses equals negative cos invisible function application 2 x minus square root of 3 sin invisible function application 2 x end style   

Selanjutnya, dapat diperhatikan ruas kanan.

begin mathsize 14px style negative k cos invisible function application open parentheses 2 x plus b close parentheses equals negative k open parentheses cos invisible function application 2 x cos invisible function application b minus sin invisible function application 2 x sin invisible function application b close parentheses equals negative k cos invisible function application 2 x cos invisible function application b plus k sin invisible function application 2 x sin invisible function application b end style  

Oleh karena itu, didapat persamaan seperti berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 sin invisible function application x open parentheses sin invisible function application x minus square root of 3 cos invisible function application x close parentheses minus 1 end cell equals cell negative k cos invisible function application open parentheses 2 x plus b close parentheses end cell row cell negative cos invisible function application 2 x minus square root of 3 sin invisible function application 2 x end cell equals cell negative k cos invisible function application 2 x cos invisible function application b plus k sin invisible function application 2 x sin invisible function application b end cell end table end style 

Dari persamaan tersebut, didapat begin mathsize 14px style 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 1 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k 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 end style dan begin mathsize 14px style 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 cell square root of 3 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 k 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 end style.

Oleh karena itu, didapat persamaan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses negative k cos invisible function application b close parentheses squared plus open parentheses k sin invisible function application b close parentheses squared end cell equals cell open parentheses negative 1 close parentheses squared plus open parentheses negative square root of 3 close parentheses squared end cell row cell k squared cos squared invisible function application b plus k squared sin squared invisible function application b end cell equals cell 1 plus 3 end cell row cell k squared open parentheses cos squared invisible function application b plus sin squared invisible function application b close parentheses end cell equals 4 row cell k squared open parentheses 1 close parentheses end cell equals 4 row cell k squared end cell equals 4 row k equals cell plus-or-minus 2 end cell end table  

Pada soal, diketahui bahwa begin mathsize 14px style k less than 0 end style, maka begin mathsize 14px style k equals negative 2 end style.

Dengan demikian, nilai dari begin mathsize 14px style k squared minus k end style adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell k squared minus k end cell equals cell open parentheses negative 2 close parentheses squared minus open parentheses negative 2 close parentheses end cell row blank equals cell 4 plus 2 end cell row blank equals 6 end table end style

Jadi, jawaban yang tepat adalah E.

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Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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