Pertanyaan

Diketahui bahwa: 3 sin 3 x + 3 cos 3 x = 3 2 , maka cos 2 2 x = . . . (SIMAK UI 2013)

Diketahui bahwa: $3 sin^{3} x + 3 cos^{3} x = 32$ , maka $cos^{2} 2 x =$ . . .

(SIMAK UI 2013)

$\frac{2}{27}$
$\frac{8}{27}$
$\frac{9}{27}$
$\frac{25}{27}$
$1$

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Habis dalam

Klaim

I. Rachmawati

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Ingat bahwa 1. sin 2 x = 2 sin x cos x ↔ 2 1 sin 2 x = sin x cos x 2. sin 2 x + cos 2 x = 1 ↔ 1 − sin 2 x = cos 2 x Diketahui 3 sin 3 x + 3 cos 3 x = 3 2 . Misal m = 3 sin 3 x dan n = 3 cos 3 x .Selanjutnya , perhatikan bahwa 3 sin 2 x + 3 cos 2 x m + n ( m + n ) 3 m 3 + n 3 + 3 mn ( m + n ) sin 2 x + cos 2 x + 3 ⋅ 3 sin 2 x ⋅ 3 cos 2 x ⋅ 3 2 1 + 3 ⋅ 3 2 sin 2 x ⋅ cos 2 x 3 ⋅ 3 2 sin 2 x ⋅ cos 2 x 3 2 sin 2 x ⋅ cos 2 x 2 sin 2 x ⋅ cos 2 x 2 ( sin x ⋅ cos x ) 2 2 ( 2 1 sin 2 x ) 2 2 ⋅ 4 1 sin 2 2 x 2 1 ( 1 − cos 2 2 x ) 1 − cos 2 2 x 1 − 27 2 27 25 = = = = = = = = = = = = = = = = 3 2 3 2 ( 3 2 ) 3 2 2 2 1 3 1 27 1 27 1 27 1 27 1 27 1 27 2 cos 2 2 x cos 2 2 x Oleh karena itu, jawaban yang benar adalah D.

Ingat bahwa

$1. sin 2 x = 2 sin x cos x \leftrightarrow \frac{1}{2} sin 2 x = sin x cos x 2. sin^{2} x + cos^{2} x = 1 \leftrightarrow 1 - sin^{2} x = cos^{2} x$

Diketahui $3 sin^{3} x + 3 cos^{3} x = 32$ . Misal $m = 3 sin^{3} x dan n = 3 cos^{3} x$ . Selanjutnya, perhatikan bahwa

$3 sin^{2} x + 3 cos^{2} x m + n (m + n)^{3} m^{3} + n^{3} + 3 mn (m + n) sin^{2} x + cos^{2} x + 3 \cdot 3 sin^{2} x \cdot 3 cos^{2} x \cdot 32 1 + 3 \cdot 3 2 sin^{2} x \cdot cos^{2} x 3 \cdot 3 2 sin^{2} x \cdot cos^{2} x 3 2 sin^{2} x \cdot cos^{2} x 2 sin^{2} x \cdot cos^{2} x 2 (sin x \cdot cos x)^{2} 2 (\frac{1}{2} sin 2 x)^{2} 2 \cdot \frac{1}{4} sin^{2} 2 x \frac{1}{2} (1 - cos^{2} 2 x) 1 - cos^{2} 2 x 1 - \frac{2}{27} \frac{25}{27} = = = = = = = = = = = = = = = = 3232 (32)^{3} 2221 \frac{1}{3} \frac{1}{27} \frac{1}{27} \frac{1}{27} \frac{1}{27} \frac{1}{27} \frac{2}{27} cos^{2} 2 x cos^{2} 2 x$