Dengan menjabarkan ruas kiri, tunjukkan bahwa: sin70∘+8cos20∘ cos40∘ cos80∘=2cos210∘

Pertanyaan

Dengan menjabarkan ruas kiri, tunjukkan bahwa:

$\sin70^\circ+8\cos20^\circ\;\cos40^\circ\;\cos80^\circ=2\cos^210^\circ$

S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Jawaban

terbukti bahwa $\sin70^\circ+8\cos20^\circ\;\cos40^\circ\;\cos80^\circ=2\cos^210^\circ$ .

Pembahasan

Ingat
rumus sudut rangkap trigonometri
$\sin\left(2\alpha\right)=2\sin\left(a\right)\;\cos\left(a\right)$ .
Rumus penjumlahan sinus
$\sin\alpha+\sin\beta=2\;\sin\frac12\left(\alpha+\beta\right)\;\cos\frac12\left(\alpha-\beta\right)\;$

Sehingga diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin 70 degree plus 8 cos 20 degree space cos 40 degree space cos 80 degree space cross times fraction numerator 2 space sin 20 degree space over denominator 2 space sin 20 degree space end fraction end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 2 space sin 20 degree space end fraction open parentheses 2 space sin 20 degree cos 20 degree close parentheses space cos 40 degree space cos 80 degree end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 2 space sin 20 degree end fraction sin 40 degree space cos 40 degree space cos 80 degree space end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 2.2 space sin 20 degree end fraction 2. space sin 40 degree space cos 40 degree space cos 80 end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 4 space sin 20 degree end fraction sin 80 degree space cos 80 degree space end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 2.4 space sin 20 degree end fraction 2. space sin 80 degree space cos 80 degree end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 8 space sin 20 degree end fraction sin 160 degree space end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 8 space sin 20 degree end fraction sin open parentheses 180 degree minus 20 degree close parentheses space end cell row blank equals cell sin 70 degree plus fraction numerator 8 over denominator 8 up diagonal strike space sin 20 end strike degree end fraction up diagonal strike sin 20 degree space end strike end cell row blank equals cell sin 70 degree plus 8 over 8 end cell row blank equals cell sin 70 degree plus 1 end cell row blank equals cell sin 70 degree plus sin 90 degree end cell row blank equals cell sin 70 degree plus sin 90 degree end cell row blank equals cell 2 sin 1 half open parentheses 70 degree plus 90 degree close parentheses space cos 1 half open parentheses 70 degree minus 90 degree close parentheses end cell row blank equals cell 2 sin 80 degree cos open parentheses negative 10 degree close parentheses end cell row blank blank cell text ingat end text space sin open parentheses 90 minus alpha close parentheses text = end text cos text α dan cos end text open parentheses negative alpha close parentheses text =cosα end text end cell row blank equals cell 2 cos 10 degree cos 10 degree end cell row blank equals cell 2 cos squared 10 degree end cell end table$

Dengan demikian, terbukti bahwa $\sin70^\circ+8\cos20^\circ\;\cos40^\circ\;\cos80^\circ=2\cos^210^\circ$ .