Pertanyaan

Buktikan bahwa: cos 3 θ cos θ − cos 6 θ cos 2 θ + cos 12 θ cos 4 θ − cos 24 θ cos 8 θ = sin 17 θ sin 15 θ

Buktikan bahwa:

$cos 3 θ cos θ - cos 6 θ cos 2 θ + cos 12 θ cos 4 θ - cos 24 θ cos 8 θ = sin 17 θ sin 15 θ$

SS

S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Pembahasan

Ingat rumus perkalian trigonometri 2 cos α cos β = cos ( α + β ) + cos ( α − β ) . Diperoleh perhitungan sebagai berikut cos 3 θ cos θ cos 6 θ cos 2 θ cos 12 θ cos 4 θ cos 24 θ cos 8 θ = = = = = = = = = = = = = = = = 2 1 ( 2 cos 3 θ cos θ ) 2 1 ( cos ( 3 θ + θ ) + cos ( 3 θ − θ ) ) 2 1 ( cos 4 θ + cos 2 θ ) 2 1 cos 4 θ + 2 1 cos 2 θ 2 1 ( 2 cos 6 θ cos 2 θ ) 2 1 ( cos ( 6 θ + 2 θ ) + cos ( 6 θ − 2 θ )) 2 1 ( cos 8 θ + cos 4 θ ) 2 1 cos 8 θ + 2 1 cos 4 θ 2 1 ( 2 cos 12 θ cos 4 θ ) 2 1 ( cos ( 12 θ + 4 θ ) + cos ( 12 θ − 4 θ )) 2 1 ( cos 16 θ + cos 8 θ ) 2 1 cos 16 θ + 2 1 cos 8 θ 2 1 ( 2 cos 24 θ cos 8 θ ) 2 1 ( cos ( 24 θ + 8 θ ) + cos ( 24 θ − 8 θ )) 2 1 ( cos 32 θ + cos 16 θ ) 2 1 cos 32 θ + 2 1 cos 16 θ Sehingga diperoleh cos 3 θ cos θ − cos 6 θ cos 2 θ + cos 12 θ cos 4 θ − cos 24 θ cos 8 θ = 2 1 cos 4 θ + 2 1 cos 2 θ − ( 2 1 cos 8 θ + 2 1 cos 4 θ ) + 2 1 cos 16 θ + 2 1 cos 8 θ − ( 2 1 cos 32 θ + 2 1 cos 16 θ ) = 2 1 cos 4 θ + 2 1 cos 2 θ − 2 1 cos 8 θ − 2 1 cos 4 θ + 2 1 cos 16 θ + 2 1 cos 8 θ − 2 1 cos 32 θ − 2 1 cos 16 θ = 2 1 cos 2 θ − 2 1 cos 8 θ + 2 1 cos 16 θ + 2 1 cos 8 θ − 2 1 cos 32 θ − 2 1 cos 16 θ = 2 1 cos 2 θ + 2 1 cos 16 θ − 2 1 cos 32 θ − 2 1 cos 16 θ = 2 1 cos 2 θ − 2 1 cos 32 θ = 2 1 ( cos 2 θ − cos 32 θ ) ingat rumus jumlah dan selisih trigonometri cos α − cos β = − 2 sin 2 1 ( α + β ) sin 2 1 ( α − β ) = = = = = = 2 1 ( cos 2 θ − cos 32 θ ) 2 1 ( − 2 sin 2 1 ( 2 θ + 32 θ ) sin 2 1 ( 2 θ − 32 θ ) ) 2 1 ( − 2 sin 2 1 ( 34 θ ) sin 2 1 ( − 30 θ ) ) 2 1 ( − 2 sin 17 θ sin ( − 15 θ ) ) − sin 17 θ sin ( − 15 θ ) − sin 17 θ − sin 15 θ sin 17 θ sin 15 θ Dengan demikian, terbukti bahwa cos 3 θ cos θ − cos 6 θ cos 2 θ + cos 12 θ cos 4 θ − cos 24 θ cos 8 θ = sin 17 θ sin 15 θ

Ingat rumus perkalian trigonometri $2 cos α cos β = cos (α + β) + cos (α - β)$ .

Diperoleh perhitungan sebagai berikut

$cos 3 θ cos θ cos 6 θ cos 2 θ cos 12 θ cos 4 θ cos 24 θ cos 8 θ = = = = = = = = = = = = = = = = \frac{1}{2} (2 cos 3 θ cos θ) \frac{1}{2} (cos (3 θ + θ) + cos (3 θ - θ)) \frac{1}{2} (cos 4 θ + cos 2 θ) \frac{1}{2} cos 4 θ + \frac{1}{2} cos 2 θ \frac{1}{2} (2 cos 6 θ cos 2 θ) \frac{1}{2} (cos (6 θ + 2 θ) + cos (6 θ - 2 θ)) \frac{1}{2} (cos 8 θ + cos 4 θ) \frac{1}{2} cos 8 θ + \frac{1}{2} cos 4 θ \frac{1}{2} (2 cos 12 θ cos 4 θ) \frac{1}{2} (cos (12 θ + 4 θ) + cos (12 θ - 4 θ)) \frac{1}{2} (cos 16 θ + cos 8 θ) \frac{1}{2} cos 16 θ + \frac{1}{2} cos 8 θ \frac{1}{2} (2 cos 24 θ cos 8 θ) \frac{1}{2} (cos (24 θ + 8 θ) + cos (24 θ - 8 θ)) \frac{1}{2} (cos 32 θ + cos 16 θ) \frac{1}{2} cos 32 θ + \frac{1}{2} cos 16 θ$

Sehingga diperoleh

$cos 3 θ cos θ - cos 6 θ cos 2 θ + cos 12 θ cos 4 θ - cos 24 θ cos 8 θ = \frac{1}{2} cos 4 θ + \frac{1}{2} cos 2 θ - (\frac{1}{2} cos 8 θ + \frac{1}{2} cos 4 θ) + \frac{1}{2} cos 16 θ + \frac{1}{2} cos 8 θ - (\frac{1}{2} cos 32 θ + \frac{1}{2} cos 16 θ) = \frac{1}{2} cos 4 θ + \frac{1}{2} cos 2 θ - \frac{1}{2} cos 8 θ - \frac{1}{2} cos 4 θ + \frac{1}{2} cos 16 θ + \frac{1}{2} cos 8 θ - \frac{1}{2} cos 32 θ - \frac{1}{2} cos 16 θ = \frac{1}{2} cos 2 θ - \frac{1}{2} cos 8 θ + \frac{1}{2} cos 16 θ + \frac{1}{2} cos 8 θ - \frac{1}{2} cos 32 θ - \frac{1}{2} cos 16 θ = \frac{1}{2} cos 2 θ + \frac{1}{2} cos 16 θ - \frac{1}{2} cos 32 θ - \frac{1}{2} cos 16 θ = \frac{1}{2} cos 2 θ - \frac{1}{2} cos 32 θ = \frac{1}{2} (cos 2 θ - cos 32 θ)$

ingat rumus jumlah dan selisih trigonometri
$cos α - cos β = - 2 sin \frac{1}{2} (α + β) sin \frac{1}{2} (α - β)$

$= = = = = = \frac{1}{2} (cos 2 θ - cos 32 θ) \frac{1}{2} (- 2 sin \frac{1}{2} (2 θ + 32 θ) sin \frac{1}{2} (2 θ - 32 θ)) \frac{1}{2} (- 2 sin \frac{1}{2} (34 θ) sin \frac{1}{2} (- 30 θ)) \frac{1}{2} (- 2 sin 17 θ sin (- 15 θ)) - sin 17 θ sin (- 15 θ) - sin 17 θ - sin 15 θ sin 17 θ sin 15 θ$