- sin 2A = 2 sin A cos A
- cos 2A = cos2 A - sin2 A
- cos 2A = 2 cos2 A - 1
- cos 2A = 1 - 2 sin2 A
- 1 + cos 2A = 2 cos2 A
- 1 - cos 2A = 2 sin2 A
- tan2A = 1−cos2A1+cos2A
- sin 2A = 2tanA1+tan2A
- cos 2A = 1−tan2A1+tan2A
- tan 2A = 2tanA1−tan2A
- sin 3A = 3 sin A - 4 sin3 A
- cos 3A = 4 cos3 A - 3 cos A
- tan 3A = 3tanA−tan3A1−3tan2A
Contoh Soal 1.
Buktikan bahwa cos 5x = 16 cos5 x - 20 cos3 x + 5 cos xJawab:
cos 5x = cos (2x + 3x)
= cos 2x cos 3x - sin 2x sin 3x
= (2 cos2 x - 1)(4 cos3 x - 3 cos x) - 2 sin x cos x (3 sin x - 4 sin3 x)
= 8 cos5 x - 10 cos3 x + 3 cos x - 6 cos x sin2 x + 8 cos x sin4 x
= 8 cos5 x - 10 cos3 x + 3 cos x - 6 cos x (1 - cos2 x) + 8 cos x (1 - cos2 x)2
= 8 cos5 x - 10 cos3 x + 3 cos x - 6 cos x + 6 cos3 x + 8 cos x - 16 cos3 x + 8 cos5 x
= 16 cos5 x - 20 cos3 x + 5 cos x
Contoh Soal 2.
Jika 13x = π, buktikan bahwa cos x cos 2x cos 3x cos 4x cos 5x cos 6x = 2-6
Jawab:
cos x cos 2x cos 3x cos 4x cos 5x cos 6x
= (12 sinx)(2 sin x cos x) cos 2x cos 3x cos 4x cos 5x cos 6x
= (12 sinx) sin 2x cos 2x cos 3x cos 4x cos 5x cos 6x
= ((12)2sinx)(2 sin 2x cos 2x) cos 3x cos 4x cos 5x cos 6x
= ((12)3 sinx)(2 sin 4x cos 4x) cos 3x cos 5x cos 6x
= ((12)3 sinx) sin 8x cos 3x cos 5x cos 6x
= ((12)4 sinx)(2 sin 5x cos 5x) cos 3x cos 6x,
[Karena, sin 8x = sin (13x - 5x) = sin (π - 5x) = sin 5x, (diberikan 13x = π)]
= ((12)4 sinx) sin 10x cos 3x cos 6x
= ((12)5 sinx) (2 sin 3x cos 3x) cos 6x,
[Karena, sin 10x = sin (13x - 3x) = sin (π - 3x) = sin 3x, (diberikan 13x = π)]
= ((12)6sinx) 2 sin 3x cos 6x
= ((12)6 sinx) sin 12x
= ((12)6 sin x) sin (13x - x)
= ((12)6 sin x) sin (π - x), [Sejak, 13x = π]
= ((12)6 sinx) sin x
= (12)6. Terbukti