Pertanyaan

Dengan menjabarkan ruas kiri, tunjukkan bahwa: sin 7 0 ∘ + 8 cos 2 0 ∘ cos 4 0 ∘ cos 8 0 ∘ = 2 cos 2 1 0 ∘

Dengan menjabarkan ruas kiri, tunjukkan bahwa:

$sin 7 0^{\circ} + 8 cos 2 0^{\circ} cos 4 0^{\circ} cos 8 0^{\circ} = 2 cos^{2} 1 0^{\circ}$

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Habis dalam

Klaim

S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Jawaban

terbukti bahwa sin 7 0 ∘ + 8 cos 2 0 ∘ cos 4 0 ∘ cos 8 0 ∘ = 2 cos 2 1 0 ∘ .

terbukti bahwa

sin 7 0^{\circ} + 8 cos 2 0^{\circ} cos 4 0^{\circ} cos 8 0^{\circ} = 2 cos^{2} 1 0^{\circ}

Pembahasan

Ingat rumus sudut rangkap trigonometri sin ( 2 α ) = 2 sin ( a ) cos ( a ) . Rumus penjumlahan sinus sin α + sin β = 2 sin 2 1 ( α + β ) cos 2 1 ( α − β ) Sehingga diperoleh = = = = = = = = = = = = = = = = sin 7 0 ∘ + 8 cos 2 0 ∘ cos 4 0 ∘ cos 8 0 ∘ × 2 s i n 2 0 ∘ 2 s i n 2 0 ∘ sin 7 0 ∘ + 2 s i n 2 0 ∘ 8 ( 2 sin 2 0 ∘ cos 2 0 ∘ ) cos 4 0 ∘ cos 8 0 ∘ sin 7 0 ∘ + 2 s i n 2 0 ∘ 8 sin 4 0 ∘ cos 4 0 ∘ cos 8 0 ∘ sin 7 0 ∘ + 2.2 s i n 2 0 ∘ 8 2. sin 4 0 ∘ cos 4 0 ∘ cos 80 sin 7 0 ∘ + 4 s i n 2 0 ∘ 8 sin 8 0 ∘ cos 8 0 ∘ sin 7 0 ∘ + 2.4 s i n 2 0 ∘ 8 2. sin 8 0 ∘ cos 8 0 ∘ sin 7 0 ∘ + 8 s i n 2 0 ∘ 8 sin 16 0 ∘ sin 7 0 ∘ + 8 s i n 2 0 ∘ 8 sin ( 18 0 ∘ − 2 0 ∘ ) sin 7 0 ∘ + 8 s i n 20 ∘ 8 sin 2 0 ∘ sin 7 0 ∘ + 8 8 sin 7 0 ∘ + 1 sin 7 0 ∘ + sin 9 0 ∘ sin 7 0 ∘ + sin 9 0 ∘ 2 sin 2 1 ( 7 0 ∘ + 9 0 ∘ ) cos 2 1 ( 7 0 ∘ − 9 0 ∘ ) 2 sin 8 0 ∘ cos ( − 1 0 ∘ ) ingat sin ( 90 − α ) = cos αdancos ( − α ) =cosα 2 cos 1 0 ∘ cos 1 0 ∘ 2 cos 2 1 0 ∘ Dengan demikian, terbukti bahwa sin 7 0 ∘ + 8 cos 2 0 ∘ cos 4 0 ∘ cos 8 0 ∘ = 2 cos 2 1 0 ∘ .

Ingat
rumus sudut rangkap trigonometri
$sin (2 α) = 2 sin (a) cos (a)$ .
Rumus penjumlahan sinus
$sin α + sin β = 2 sin \frac{1}{2} (α + β) cos \frac{1}{2} (α - β)$

Sehingga diperoleh

$= = = = = = = = = = = = = = = = sin 7 0^{\circ} + 8 cos 2 0^{\circ} cos 4 0^{\circ} cos 8 0^{\circ} \times \frac{2 s i n 2 0 ^{\circ}}{2 s i n 2 0 ^{\circ}} sin 7 0^{\circ} + \frac{8}{2 s i n 2 0 ^{\circ}} (2 sin 2 0^{\circ} cos 2 0^{\circ}) cos 4 0^{\circ} cos 8 0^{\circ} sin 7 0^{\circ} + \frac{8}{2 s i n 2 0 ^{\circ}} sin 4 0^{\circ} cos 4 0^{\circ} cos 8 0^{\circ} sin 7 0^{\circ} + \frac{8}{2.2 s i n 2 0 ^{\circ}} 2. sin 4 0^{\circ} cos 4 0^{\circ} cos 80 sin 7 0^{\circ} + \frac{8}{4 s i n 2 0 ^{\circ}} sin 8 0^{\circ} cos 8 0^{\circ} sin 7 0^{\circ} + \frac{8}{2.4 s i n 2 0 ^{\circ}} 2. sin 8 0^{\circ} cos 8 0^{\circ} sin 7 0^{\circ} + \frac{8}{8 s i n 2 0 ^{\circ}} sin 16 0^{\circ} sin 7 0^{\circ} + \frac{8}{8 s i n 2 0 ^{\circ}} sin (18 0^{\circ} - 2 0^{\circ}) sin 7 0^{\circ} + \frac{8}{8 s i n 20 ^{\circ}} sin 2 0^{\circ} sin 7 0^{\circ} + \frac{8}{8} sin 7 0^{\circ} + 1 sin 7 0^{\circ} + sin 9 0^{\circ} sin 7 0^{\circ} + sin 9 0^{\circ} 2 sin \frac{1}{2} (7 0^{\circ} + 9 0^{\circ}) cos \frac{1}{2} (7 0^{\circ} - 9 0^{\circ}) 2 sin 8 0^{\circ} cos (- 1 0^{\circ}) ingat sin (90 - α) = cos α dan cos (- α) =cosα 2 cos 1 0^{\circ} cos 1 0^{\circ} 2 cos^{2} 1 0^{\circ}$