Pertanyaan

13. Hitunglah tanpa menggunakan kalkulator atau tabel trigonometri. f. sin 10 5 ∘ − sin 1 5 ∘ cos 22 5 ∘ − cos 19 5 ∘

13. Hitunglah tanpa menggunakan kalkulator atau tabel trigonometri.

f. $\frac{cos 22 5 ^{\circ} - cos 19 5 ^{\circ}}{sin 10 5 ^{\circ} - sin 1 5 ^{\circ}}$

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P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

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

Pembahasan

Ingat rumus identitas selisih cosinus dan selisih sinus berikut: Dari soal diketahui: x = y = a = b = Sehingga persamaan trigonometri tersebut menjadi: Kita cari nilai dengan mengaplikasikan rumus selisih sudut pada sinus, diperoleh: Selanjutnya, kita substitusi nilai ke persamaan sebelumnya, diperoleh: Dengan demikian, .

Ingat rumus identitas selisih cosinus dan selisih sinus berikut:

Dari soal diketahui:

x = 225 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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