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Dengan menggunakan rumus cosinus selisih dua sudut, nilai dari  adalah ....

Pertanyaan

Dengan menggunakan rumus cosinus selisih dua sudut, nilai dari begin mathsize 14px style sin space 15 degree end style adalah ....

Pembahasan Soal:

Dengan konsep silisih sudut pada sinus.

begin mathsize 14px style Sin space left parenthesis straight alpha minus straight beta right parenthesis equals sinα times cosβ minus sinβ times cosα end style 

untuk begin mathsize 14px style 15 degree end style kita ubah menjadi dua sudut istimewa yaitu begin mathsize 14px style 45 degree minus 30 degree end style sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell Sin space 15 degree end cell equals cell left parenthesis 45 degree minus 30 degree right parenthesis end cell row blank equals cell sin 45 degree times cos 30 degree minus sin 30 degree times cos 45 degree end cell row blank equals cell 1 half square root of 2 times 1 half square root of 3 minus 1 half times 1 half square root of 2 end cell row blank equals cell 1 fourth square root of 6 minus 1 fourth square root of 2 end cell row blank equals cell 1 fourth left parenthesis square root of 6 minus square root of 2 right parenthesis end cell end table end style 

Jadi, Nilai dari 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 space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 15 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree 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 cell 1 fourth left parenthesis square root of 6 minus square root of 2 right parenthesis end cell end table end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. RGFLLIMA

Terakhir diupdate 30 April 2021

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Pembahasan Soal:

Ingat: sin space open parentheses A plus-or-minus B close parentheses equals sin space A space cos space B space plus-or-minus space cos space A space sin space B 

          cos space A space cos space B equals 1 half cos space open parentheses A plus B close parentheses plus 1 half cos space open parentheses A minus B close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 52 comma 5 degree minus tan space 7 comma 5 degree over denominator tan space 165 degree plus tan space 75 degree end fraction end cell equals cell fraction numerator begin display style fraction numerator sin space 52 comma 5 degree over denominator cos space 52 comma 5 degree end fraction end style minus begin display style fraction numerator sin space 7 comma 5 degree over denominator cos space 7 comma 5 degree end fraction end style over denominator begin display style fraction numerator sin space 165 degree over denominator cos space 165 degree end fraction end style plus begin display style fraction numerator sin space 75 degree over denominator cos space 75 degree end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space 52 comma 5 degree space cos space 7 comma 5 degree minus cos space 52 comma 5 degree sin space 7 comma 5 degree over denominator cos space 52 comma 5 degree space cos space 7 comma 5 degree end fraction end style over denominator begin display style fraction numerator sin space 165 degree space cos space 75 degree plus cos space 165 degree sin space 75 degree over denominator cos space 165 degree cos space 75 degree end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space open parentheses 52 comma 5 degree minus 7 comma 5 degree close parentheses over denominator begin display style 1 half end style cos space open parentheses 52 comma 5 degree plus 7 comma 5 degree close parentheses plus begin display style 1 half end style cos space open parentheses 52 comma 5 degree minus 7 comma 5 degree close parentheses end fraction end style over denominator begin display style fraction numerator sin space open parentheses 165 degree plus 75 degree close parentheses over denominator 1 half cos space open parentheses 165 degree plus 75 degree close parentheses plus 1 half cos space open parentheses 165 degree minus 75 degree close parentheses end fraction end style end fraction cross times fraction numerator begin display style 1 half end style over denominator begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space 45 degree over denominator cos space 60 degree plus cos space 45 degree end fraction end style space over denominator begin display style fraction numerator sin space 240 degree over denominator cos space 240 degree plus cos space 90 degree end fraction end style end fraction end cell end table

Dengan sudut berelasi

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 240 degree end cell equals cell sin space open parentheses 180 degree plus 60 degree close parentheses end cell row blank equals cell negative sin space 60 degree end cell row blank equals cell negative 1 half square root of 3 end cell end table

dan 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 240 degree end cell equals cell cos space open parentheses 180 degree plus 60 degree close parentheses end cell row blank equals cell negative cos space 60 degree end cell row blank equals cell negative 1 half end cell end table

maka menjadi

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 52 comma 5 degree minus tan space 7 comma 5 degree over denominator tan space 165 degree plus tan space 75 degree end fraction end cell equals cell fraction numerator begin display style 1 half end style square root of 2 over denominator begin display style 1 half end style plus begin display style 1 half end style square root of 2 end fraction cross times fraction numerator negative begin display style 1 half end style plus 0 over denominator negative begin display style 1 half end style square root of 3 end fraction cross times 4 over 4 end cell row blank equals cell fraction numerator square root of 2 over denominator square root of 6 minus square root of 3 end fraction cross times fraction numerator square root of 6 minus square root of 3 over denominator square root of 6 minus square root of 3 end fraction end cell row blank equals cell fraction numerator 2 square root of 3 minus square root of 6 over denominator 6 minus 3 end fraction end cell row blank equals cell 1 third open parentheses 2 square root of 3 minus square root of 6 close parentheses end cell end table

Jadi, jawaban yang tepat adalah A

0

Roboguru

Untuk menentukan tinggi bukit (), seorang pengamat mengukur sudut elevasi ( dan ) dari titik A dan B yang berjarak . Tunjukkan: b.

Pembahasan Soal:

Nilai tangen diperoleh dari perbandingan sisi depan sudut dengan sisi samping sudut pada segitiga siku-siku.

Ingat!

tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction 

Selisih Dua Sudut pada Sinus

sin space open parentheses alpha minus beta close parentheses equals sin space alpha space cos space beta minus cos space alpha space sin space beta 

Misalkan sisi samping alpha adalah x.

Berdasarkan gambar di atas, diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space alpha end cell equals cell t over x end cell row cell tan space beta end cell equals cell fraction numerator t over denominator d plus x end fraction end cell end table 

Maka, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space alpha end cell equals cell t over x end cell row x equals cell fraction numerator t over denominator tan space alpha end fraction space... space left parenthesis straight i right parenthesis end cell row blank blank blank row cell tan space beta end cell equals cell fraction numerator t over denominator d plus x end fraction end cell row cell d plus x end cell equals cell fraction numerator t over denominator tan space beta end fraction end cell row x equals cell fraction numerator t over denominator tan space beta end fraction minus d space... space left parenthesis ii right parenthesis end cell end table 

Terdapat kesalahan pada soal. Bukan t equals fraction numerator d space sin space alpha times cos space beta over denominator sin space open parentheses alpha minus beta close parentheses end fraction, melainkan table attributes columnalign right center left columnspacing 0px end attributes row blank blank t 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 cell fraction numerator d space sin space alpha space sin space beta over denominator sin space open parentheses alpha space minus beta close parentheses end fraction end cell end table.

Akan ditunjukkan table attributes columnalign right center left columnspacing 0px end attributes row blank blank t 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 cell fraction numerator d space sin space alpha space sin space beta over denominator sin space open parentheses alpha space minus beta close parentheses end fraction end cell end table.

Substitusikan persamaan (i) dan (ii).

table attributes columnalign right center left columnspacing 0px end attributes row x equals x row cell fraction numerator t over denominator tan space alpha end fraction end cell equals cell fraction numerator t over denominator tan space beta end fraction minus d end cell row cell fraction numerator t over denominator tan space alpha end fraction times tan space alpha space tan space beta end cell equals cell open parentheses fraction numerator t over denominator tan space beta end fraction minus d close parentheses times tan space alpha space tan space beta end cell row cell t space tan space beta end cell equals cell t space tan space alpha minus d space tan space alpha space tan space beta end cell row cell d space tan space alpha space tan space beta end cell equals cell t space tan space alpha minus t space tan space beta end cell row cell d space tan space alpha space tan space beta end cell equals cell t space open parentheses tan space alpha minus tan space beta close parentheses end cell row cell fraction numerator d space tan space alpha space tan space beta over denominator tan space alpha minus tan space beta end fraction end cell equals t row t equals cell fraction numerator d space tan space alpha space tan space beta over denominator tan space alpha minus tan space beta end fraction end cell row t equals cell fraction numerator begin display style fraction numerator d space sin space alpha space sin space beta over denominator cos space alpha space cos space beta end fraction end style over denominator begin display style fraction numerator sin space alpha over denominator cos space alpha end fraction end style minus fraction numerator sin space beta over denominator cos space beta end fraction end fraction end cell row t equals cell fraction numerator begin display style fraction numerator d space sin space alpha space sin space beta over denominator up diagonal strike cos space alpha space cos space beta end strike end fraction end style over denominator begin display style fraction numerator sin space alpha space cos space beta minus cos space alpha space sin space beta over denominator up diagonal strike cos space alpha space cos space beta end strike end fraction end style end fraction end cell row t equals cell fraction numerator d space sin space alpha space sin space beta over denominator sin space alpha space cos space beta minus cos space alpha space sin space beta end fraction end cell row t equals cell fraction numerator d space sin space alpha space sin space beta over denominator sin space open parentheses alpha space minus beta close parentheses end fraction end cell end table  

Jadi, terbukti bahwa table attributes columnalign right center left columnspacing 0px end attributes row blank blank t 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 cell fraction numerator d space sin space alpha space sin space beta over denominator sin space open parentheses alpha space minus beta close parentheses end fraction end cell end table.

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Roboguru

Himpunan penyelesaian dari persamaan  untuk  adalah ....

Pembahasan Soal:

Ingat rumus selisih sinus berikut untuk penyelesaian soal tersebut, yaitu:

sin open parentheses A plus B close parentheses minus sin open parentheses A minus B close parentheses equals 2 space cos space A space sin space B 

Dari soal diketahui bahwa A = x dan B = 15degree. Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses x plus 15 close parentheses degree minus sin space open parentheses x minus 15 close parentheses degree end cell equals cell 1 half open parentheses square root of 3 minus 1 close parentheses end cell row cell 2 space cos space x space sin space 15 degree end cell equals cell 1 half open parentheses square root of 3 minus 1 close parentheses end cell end table 

Kita akan cari nilai dari sin 15 degree yaitu sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 15 degree end cell equals cell sin open parentheses 45 minus 30 close parentheses degree end cell row blank equals cell sin space 45 degree space cos space 30 degree minus cos space 45 degree plus sin space 30 degree end cell row blank equals cell open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses minus open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses open parentheses 1 half close parentheses end cell row blank equals cell fraction numerator square root of 2 over denominator 4 end fraction open parentheses square root of 3 minus 1 close parentheses end cell end table 

Selanjutnya, substitusi nilai sin 15 degree, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space cos space x space sin space 15 degree end cell equals cell 1 half open parentheses square root of 3 minus 1 close parentheses end cell row cell 2 space cos space x times fraction numerator square root of 2 over denominator 4 end fraction up diagonal strike open parentheses square root of 3 minus 1 close parentheses end strike end cell equals cell 1 half up diagonal strike open parentheses square root of 3 minus 1 close parentheses end strike end cell row cell cos space x times fraction numerator square root of 2 over denominator up diagonal strike 2 end fraction end cell equals cell fraction numerator 1 over denominator up diagonal strike 2 end fraction end cell row cell cos space x end cell equals cell fraction numerator 1 over denominator square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row cell cos space x end cell equals cell 1 half square root of 2 end cell row cell cos space x end cell equals cell cos space 45 degree end cell row blank blank blank end table 

Berdasarkan rumus dasar trigonometri untuk cosinus yaitu:

cos space x equals cos space A left right double arrow x equals A plus n times 360 space atau space x equals negative straight A plus n times 360 

Maka: x equals 45 plus n times 360 atau x equals negative 45 plus n times 360.

table row cell n equals 0 end cell rightwards double arrow cell x subscript 1 equals 45 plus 0 equals 45 end cell row blank rightwards double arrow cell x subscript 2 equals negative 45 plus 0 equals negative 45 space open parentheses TM close parentheses end cell row cell n equals 1 end cell rightwards double arrow cell x subscript 3 equals 45 plus 360 equals 405 space open parentheses TM close parentheses end cell row blank blank cell x subscript 4 equals negative 45 plus 360 equals 315 end cell end table 

Dengan demikian, himpunan penyelesaian adalah HP = open curly brackets 45 degree comma space 315 degree close curly brackets.

Oleh karena itu, jawaban yang benar adalah B.

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Roboguru

Nilai dari  adalah ....

Pembahasan Soal:

sin open parentheses 15 degree close parentheses plus cos open parentheses 105 degree close parentheses minus 2 sin open parentheses 37 degree close parentheses plus cos open parentheses 60 degree close parentheses plus 2 cos open parentheses 53 degree close parentheses 

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses 15 degree close parentheses end cell equals cell sin open parentheses 60 degree minus 45 degree close parentheses end cell row blank equals cell sin open parentheses 60 degree close parentheses cos open parentheses 45 degree close parentheses minus cos open parentheses 60 degree close parentheses sin open parentheses 45 degree close parentheses end cell row blank equals cell 1 half square root of 3 times 1 half square root of 2 minus 1 half times 1 half square root of 2 end cell row blank equals cell 1 fourth square root of 6 minus 1 fourth square root of 2 end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos open parentheses 105 degree close parentheses end cell equals cell cos open parentheses 60 degree plus 45 degree close parentheses end cell row blank equals cell cos open parentheses 60 degree close parentheses cos open parentheses 45 degree close parentheses minus sin open parentheses 60 degree close parentheses sin open parentheses 45 degree close parentheses end cell row blank equals cell 1 half times 1 half square root of 2 minus 1 half square root of 3 times 1 half square root of 2 end cell row blank equals cell 1 fourth square root of 2 minus 1 fourth square root of 6 end cell end table 

sin open parentheses 37 degree close parentheses equals cos open parentheses 53 degree close parentheses equals 3 over 5 

sin open parentheses 15 degree close parentheses plus cos open parentheses 105 degree close parentheses minus 2 sin open parentheses 37 degree close parentheses plus cos open parentheses 60 degree close parentheses plus 2 cos open parentheses 53 degree close parentheses equals 1 fourth square root of 6 minus 1 fourth square root of 2 plus 1 fourth square root of 2 minus 1 fourth square root of 6 minus 2 open parentheses 3 over 5 close parentheses plus 1 half plus 2 open parentheses 3 over 5 close parentheses equals 1 half 

Maka, pilihan jawaban yang benar adalah C.

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Roboguru

13. Hitunglah tanpa menggunakan kalkulator atau tabel trigonometri. f.

Pembahasan Soal:

Ingat rumus identitas selisih cosinus dan selisih 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:

x225 degree 

y = 195 degree 

a = 105 degree 

b = 15 degree 

Sehingga persamaan trigonometri tersebut menjadi:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos space 225 degree minus cos space 195 degree over denominator sin space 105 degree minus sin 15 degree end fraction end cell equals cell fraction numerator negative 2 space sin space 1 half open parentheses 225 degree plus 195 degree close parentheses space sin space 1 half open parentheses 225 degree minus 195 degree close parentheses over denominator 2 space cos space 1 half open parentheses 105 degree plus 15 degree close parentheses space sin space 1 half open parentheses 105 degree minus 15 degree close parentheses end fraction end cell row blank equals cell fraction numerator negative 2 space sin space 1 half open parentheses 420 degree close parentheses space sin space 1 half open parentheses 30 degree close parentheses over denominator 2 space cos space 1 half open parentheses 120 degree close parentheses space sin space 1 half open parentheses 90 degree close parentheses end fraction end cell row blank equals cell fraction numerator negative 2 space sin space 210 degree space sin space 15 degree over denominator 2 space cos space 60 degree space sin space 45 degree end fraction end cell row blank equals cell fraction numerator negative 2 open parentheses negative begin display style 1 half end style close parentheses open parentheses sin space 15 degree close parentheses over denominator 2 open parentheses begin display style 1 half end style close parentheses open parentheses begin display style 1 half end style square root of 2 close parentheses end fraction end cell row blank equals cell fraction numerator sin space 15 degree over denominator begin display style 1 half end style square root of 2 end fraction end cell end table end style 

Kita cari nilai sin space 15 degree dengan mengaplikasikan rumus selisih sudut pada sinus, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 15 degree end cell equals cell sin space open parentheses 45 degree minus 30 degree close parentheses end cell row blank equals cell sin space 45 degree space cos space 30 degree minus cos space 45 degree space sin space 30 degree 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 6 minus 1 fourth square root of 2 end cell row blank equals cell fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction end cell end table 

Selanjutnya, kita substitusi nilai sin space 15 degree ke persamaan sebelumnya, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos space 225 degree minus cos space 195 degree over denominator sin space 105 degree minus sin 15 degree end fraction end cell equals cell fraction numerator sin space 15 degree over denominator begin display style 1 half end style square root of 2 end fraction end cell row blank equals cell fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction cross times fraction numerator 2 over denominator square root of 2 end fraction end cell row blank equals cell fraction numerator 2 open parentheses square root of 6 minus square root of 2 close parentheses over denominator 4 square root of 2 end fraction end cell row blank equals cell 1 half open parentheses fraction numerator square root of 6 minus square root of 2 over denominator square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction close parentheses end cell row blank equals cell 1 half open parentheses fraction numerator square root of 12 minus 2 over denominator 2 end fraction close parentheses end cell row blank equals cell 1 half open parentheses fraction numerator 2 square root of 3 minus 2 over denominator 2 end fraction close parentheses end cell row blank equals cell 1 half open parentheses fraction numerator up diagonal strike 2 open parentheses square root of 3 minus 1 close parentheses over denominator up diagonal strike 2 end fraction close parentheses end cell row blank equals cell 1 half open parentheses square root of 3 minus 1 close parentheses end cell end table 

Dengan demikian, fraction numerator cos space 225 degree minus cos space 195 degree over denominator sin space 105 degree minus sin 15 degree end fraction equals 1 half open parentheses square root of 3 minus 1 close parentheses.

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