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Buktikanlah setiap identitas berikut. cos3α−cosαsin3α−sinα​=−cotan2α

Pertanyaan

Buktikanlah setiap identitas berikut.

cos3αcosαsin3αsinα=cotan2α

 

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

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

cosAcosB=2sin21(A+B)sin21(AB)

Dengan menggunakan konsep di atas, diperoleh hasil:

cos3αcosαsin3αsinα=====2sin21(3α+α)sin21(3αα)2cos21(3α+α)sin21(3αα)sin21(4α)sin21(2α)cos21(4α)sin21(2α)sin2αsinαcos2αsinαsin2αcos2αcotan2α

Jadi, terbukti bahwa cos3αcosαsin3αsinα=cotan2α.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Rajib

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 13 September 2021

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Pertanyaan yang serupa

Nilai dari sin70∘−sin20∘cos20∘−cos70∘​ adalah ... .

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

cosAcosB=2sin21(A+B)sin21(AB)

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

Dengan menggunakan konsep di atas, diperoleh hasil:

sin70sin20cos20cos70========2cos21(70+20)sin21(7020)2sin21(20+70)sin21(2070)2cos21(90)sin21(50)2sin21(90)sin21(50)2cos45sin252sin45sin(25)2cos45sin252sin45(sin25)2cos45sin252sin45sin25cos45sin45tan451

Jadi, sin70sin20cos20cos70=1.

Oleh karena itu, jawaban yang benar adalah B.

Roboguru

16. Buktikan. b.

Pembahasan Soal:

Ingat rumus identitas selisih cosinus dan sinus berikut:

  • cos space x minus cos space y equals negative 2 space sin space 1 half open parentheses x plus y close parentheses space sin space 1 half open parentheses x minus y close parentheses 
  • sin space a minus sin space b equals 2 space cos space 1 half open parentheses a plus b close parentheses space sin space 1 half open parentheses a minus b close parentheses  

Dari soal diketahui:

x = 3x 

y = 5x 

a = 3x 

b = x 

Sehingga persamaan trigonometri tersebut menjadi:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos space 3 x minus cos space 5 x over denominator sin space 3 x minus sin space x end fraction end cell equals cell fraction numerator negative 2 space sin space begin display style 1 half end style open parentheses 3 x plus 5 x close parentheses space sin space begin display style 1 half end style open parentheses 3 x minus 5 x close parentheses over denominator 2 space cos space begin display style 1 half end style open parentheses 3 x plus x close parentheses space sin space begin display style 1 half end style open parentheses 3 x minus x close parentheses end fraction end cell row blank equals cell negative fraction numerator sin space begin display style 1 half end style open parentheses 8 x close parentheses space sin space begin display style 1 half end style open parentheses negative 2 x close parentheses over denominator cos space begin display style 1 half end style open parentheses 4 x close parentheses space sin space begin display style 1 half end style open parentheses 2 x close parentheses end fraction end cell row blank equals cell negative fraction numerator sin space 4 x space sin space open parentheses negative x close parentheses over denominator cos space 2 x space sin space open parentheses x close parentheses end fraction end cell row blank equals cell negative open parentheses negative fraction numerator sin space 4 x space up diagonal strike sin space open parentheses x close parentheses end strike over denominator cos space 2 x up diagonal strike space sin space open parentheses x close parentheses end strike end fraction close parentheses end cell row blank equals cell fraction numerator sin space 4 x over denominator cos space 2 x end fraction end cell end table 

Ingat pula rumus trigonometri sudut ganda, dimana sin space 2 x equals 2 space sin space x space cos space x. Sehingga, sin space 4 x dapat diubah menjadi 2 space sin space 2 x space cos space 2 x.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos space 3 x minus cos space 5 x over denominator sin space 3 x minus sin space x end fraction end cell equals cell fraction numerator sin space 4 x over denominator cos space 2 x end fraction end cell row blank equals cell fraction numerator 2 space sin space 2 x space up diagonal strike cos space 2 x end strike over denominator up diagonal strike cos space 2 x end strike end fraction end cell row blank equals cell 2 space sin space 2 x end cell end table 

Dengan demikian,

terbukti bahwa fraction numerator cos space 3 x minus cos space 5 x over denominator sin space 3 x minus sin space x end fraction equals 2 space sin space 2 x.

Roboguru

Nilai dari  adalah ....

Pembahasan Soal:

begin mathsize 14px style bold i bold right parenthesis bold space Jabarkan space bentuk space sin invisible function application space straight alpha minus sin space invisible function application straight beta space dan space cos space invisible function application straight alpha minus cos space invisible function application straight beta  sin invisible function application space straight alpha minus sin invisible function application space straight beta equals 2 space cos invisible function application 1 half open parentheses straight alpha plus straight beta close parentheses space sin invisible function application 1 half open parentheses straight alpha minus straight beta close parentheses  cos invisible function application space straight alpha minus cos invisible function application space straight beta equals negative 2 space sin invisible function application 1 half open parentheses straight alpha plus straight beta close parentheses space sin invisible function application 1 half open parentheses straight alpha minus straight beta close parentheses  Sehingga  sin space invisible function application 40 degree minus sin invisible function application space 20 degree  equals 2 space cos invisible function application space 1 half open parentheses 40 degree plus 20 degree close parentheses space sin invisible function application 1 half open parentheses 40 degree minus 20 degree close parentheses  equals 2 space cos space invisible function application 30 degree space sin space invisible function application 10 degree  equals 2 open parentheses 1 half square root of 3 close parentheses space sin space invisible function application 10 degree equals square root of 3 space sin space invisible function application 10 degree  dan  cos invisible function application space 40 degree minus cos space invisible function application 20 degree  equals negative 2 space sin invisible function application 1 half open parentheses 40 degree plus 20 degree close parentheses space sin invisible function application 1 half open parentheses 40 degree minus 20 degree close parentheses  equals negative 2 space sin space invisible function application 30 degree space sin space invisible function application 10 degree  equals negative 2 open parentheses 1 half close parentheses space sin space invisible function application 10 degree  equals negative sin space invisible function application 10 degree    bold ii bold right parenthesis bold space Substitusi space ke space soal  fraction numerator sin invisible function application space 40 degree minus sin space invisible function application 20 degree over denominator cos space invisible function application 40 degree minus cos space invisible function application 20 degree end fraction equals fraction numerator square root of 3 space sin space invisible function application 10 degree over denominator negative sin invisible function application space 10 degree end fraction equals negative square root of 3 end style

Roboguru

Nilai dari  adalah ...

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space alpha minus sin space beta end cell equals cell 2 cos space 1 half open parentheses alpha plus beta close parentheses times sin space 1 half open parentheses alpha minus beta close parentheses end cell row cell cos space alpha minus cos space beta end cell equals cell negative 2 sin space 1 half open parentheses alpha plus beta close parentheses times sin space 1 half open parentheses alpha minus beta close parentheses end cell end table 

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 280 degree minus sin space 20 degree over denominator cos space 340 degree minus cos space 80 degree end fraction end cell row blank equals cell fraction numerator 2 cos space 1 half open parentheses 280 degree plus 20 degree close parentheses times sin space 1 half open parentheses 280 degree minus 20 degree close parentheses over denominator negative 2 sin space 1 half open parentheses 340 degree plus 80 degree close parentheses times sin space 1 half open parentheses 340 degree minus 80 degree close parentheses end fraction end cell row blank equals cell fraction numerator 2 cos space begin display style fraction numerator 300 degree over denominator 2 end fraction end style times sin space begin display style fraction numerator 260 degree over denominator 2 end fraction end style over denominator negative 2 sin space begin display style fraction numerator 420 degree over denominator 2 end fraction end style times sin space begin display style fraction numerator 260 degree over denominator 2 end fraction end style end fraction end cell row blank equals cell negative fraction numerator cos space begin display style 150 degree end style over denominator sin space begin display style 210 degree end style end fraction end cell row blank equals cell negative fraction numerator cos space open parentheses 180 degree minus 30 degree close parentheses over denominator sin space begin display style open parentheses 180 degree plus 30 degree close parentheses end style end fraction end cell row blank equals cell negative fraction numerator negative cos space begin display style 30 degree end style over denominator negative sin space begin display style 30 degree end style end fraction end cell row blank equals cell negative fraction numerator cos space begin display style 30 degree end style over denominator sin space begin display style 30 degree end style end fraction end cell row blank equals cell negative fraction numerator begin display style 1 half square root of 3 end style over denominator begin display style 1 half end style end fraction end cell row blank equals cell negative square root of 3 end cell end table  

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 280 degree minus sin space 20 degree over denominator cos space 340 degree minus cos space 80 degree end fraction 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 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.

Roboguru

Pembahasan Soal:

sin space A space minus space sin space B space equals space 2 space cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses  cos space A space minus space cos space B space equals space minus 2 space sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses    fraction numerator sin space 105 degree space minus space sin space 15 degree over denominator cos space 75 degree space minus space cos space 15 degree end fraction space equals space fraction numerator 2 space cos open parentheses begin display style fraction numerator 105 degree thin space plus space 15 degree over denominator 2 end fraction end style close parentheses space sin open parentheses begin display style fraction numerator 105 degree minus 15 degree over denominator 2 end fraction end style close parentheses over denominator negative 2 space sin open parentheses begin display style fraction numerator 75 degree plus 15 degree over denominator 2 end fraction end style close parentheses space sin open parentheses begin display style fraction numerator 75 degree minus 15 degree over denominator 2 end fraction end style close parentheses 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 equals space fraction numerator 2 space cos space 60 space sin space 45 over denominator negative 2 space sin space 45 space sin space 30 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 equals space minus fraction numerator cos space 60 over denominator sin space 30 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 equals space minus fraction numerator begin display style 1 half end style over denominator begin display style 1 half end style 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 equals space minus 1

Roboguru

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