1. 360° = 2π radian
π radian = 360°/2 = 180°
Jadi, π radian = 180°
2. 1° = π /180° radian
= 3,14159/180 radian
Jadi, 1° = 0,02 radian
3. 1 radian = 180°/π
= 180/3,14159
Jadi, 1° radian = 57,296° atau 57,3°
1. Sin 0° = 0 Sin 30° = 1/2
Sin 45° = 1/2 √2
Sin 60° = 1/2 √3
Sin 90° = 1
2. Cos 0° = 1
Cos 30° = 1/2 √3
Cos 45° = 1/2 √2
Cos 60° = 1/2
Cos 90° = 0
3. Tan 0° = 0
Tan 30° = 1/3 √3
Tan 45° = 1
Tan 60° = √3
Tan 90° = ∞
4. Cosc A = 1/sin A
Sec A = 1/Cos A
Cotg A = 1/Tg ARumus2 :
Kuadran II = (180° - α)Kuadran III = (180° + α)
Kuadran IV = (360° - α)Untuk 0° < α < 90°
Contoh Soal :
= Sin 30°
= 1/2
2. Cos 120° = Cos (180° - 60°)
= - Cos 60°
= -1/2
3. Tan 315° = Tan (360° - 45°)
= - Tan 45°
= -1
Jadi, Sin (-α) = - Sin α
Tan (-α) = - Tan α
Cos (-α) = Cos α Contoh Soal :
1. Sin (-960°) = - Sin 960° = - Sin 240°
= - Sin (180° + 60°)
= - Sin (-Sin 60°)
= 1/2 √32. Tan (-1395°) = - Tan 1395
= - Tan 315
= - Tan (360° - 45°)
= - Tan (-Tan 45°)
= 1
3. Cos (-600°) = Cos 600°
= Cos 240°
= Cs (180° + 60°)
= - Cos 60°
= -1/2
* PERSAMAAN TRIGONOMETRI *
1. f : x => ax + b (Pemetaan)
2. f(x) = ax + b (Rumus)
3. f = ax + b (Persamaan)
Contoh Soal :
1. f : x => 2 Sin x + cos 2x
f(45°) = ...
Jawaban :
f(x) = 2 sin x + cos 2x
f(45°) = 2 sin (45°) + cos 2(45°)
= 2 sin 45° + cos 90°
= 2 . 1/2√2 + 0
= √2
* PERBANDINGAN TRIGONOMETRI *
1. Sin A = y/r 4. Cosc A = r/y
2. Cos A = x/r 5. Sec A = r/x
3. Tan A = y/x 6. Cotg A = x/y
Contoh Soal :
1. Tentukan HP dr 2 sin x = 1 , 0° < x < 360°
Jawaban :
* PERSAMAAN TRIGONOMETRI *
A. Sin x = Sin a
x1 = a + k . 360
x2 = (180 - a) + k . 360
k € Bilangan Bulat
k = { ... , ... , -2 , -1 , 0 , 1 , 2 , ... }
Contoh Soal :
1. Tentukan HP dr 2 sin x = 1 , 0° < x < 360°
Jawaban :
* 2 Sin x = 1
Sin x = 1/2 (kuadran I dan II)
Sin x = Sin 30°
x = 30° + k . 360°
k = 0 => x1 = 30° + 0 . 360° = 30°
=> x2 = (180° - 30°) + 0 . 360° = 150°
Jadi, HP {30° , 150°}
2.Tentukan HP dr 2 sin 2x = -√3 , 0° < x < 360°
Jawaban :
* 2 Sin 2x = -√3
Sin 2x = -√3/2 (kuadran III dan IV)
Sin 2x = Sin (180° + 60°)
Sin 2x = Sin 240°
2x = 240° + k . 360°
x1 = 120° + k . 180°
k = 0 => x1 = 120° + 0 .180° = 120°
k = 1 => x1 = 120° + 1 . 180° = 300°
2x = (180° - 240°) + k . 360°
2x = - 60° + k . 360°
x2 = - 30° + k . 180°
k = 1 => x2 = - 30° + 1 . 180° = 150°
k = 2 => x2 = - 30° + 2 . 180° = 330°
Jadi, HP {120° , 150° , 300° , 330°}
B. Cos x = Cos a
x1 = a + k . 360°
x2 = - a + k . 360°
k € Bilangan Bulat
k = { ... , ... , -2 , -1 , 0 , 1 , 2 , ... }
Contoh Soal :
1. Tentukan HP dr cos x = -1/2 , 0° < x < 360°
Jawaban :
* Cos x = -1/2 (kuadran II dan III)
Cos x = Cos (180° - 60°)
Cos x = Cos 120
x1 = 120° + k . 360°
k = 0 => x = 120° + 0 . 360° = 120°
x2 = - 120° + k . 360°
k = 1 => x = - 120° + 1 . 360° = 240°
Jadi, HP {120 , 240}
2. Tentukan HP dr 2 cos 3x = - √3 , 0° < x < 360°
Jawaban :
Jadi, HP {120 , 240}
2. Tentukan HP dr 2 cos 3x = - √3 , 0° < x < 360°
Jawaban :
* 2 cos 3x = - √3
cos 3x = - √3/2 (kuadran II dan III)
cos 3x = (180° - 30° )
cos 3x = 150°
3x = 150° + k . 360°
x1 = 50° + k . 120°
k = 0 => x1 = 50° + 0 . 120° = 50°
k = 1 => x1 = 50° + 1 . 120° = 170°
x2 = - 50° + k . 120°
k = 1 => x2 = - 50° + 1 . 120° = 70°
Jadi, HP {50° , 70° , 170°}
C. Tan x = tan a
x = a + k . 180°
k € Bilangan Bulat
k = { ... , ... , -2 , -1 , 0 , 1 , 2 , ... }
Contoh Soal :
1. Tentukan HP dr tan 2x = - √3/3 , 0° < x < 360° Jawaban :
* tan 2x = - √3/3 (kuadran II dan IV)
tan 2x = (180° - 30°)
tan 2x = 150°
2x = 150° + k . 180°
x1 = 75° + k . 90°
k = 0 => x1 = 75° + 0 . 90° = 75°
k = 1 => x1 = 75° + 1 . 90° = 165°
Jadi, HP {75° , 165°}
r = √(x2 + y2)
Tan α = y/x
Rumus2 :
* Mengubah Koordinat Kartesius ke Koordinat Kutub *
r = √(x2 + y2)
Tan α = y/xRumus2 :
1. II = 180° - α
2. III = 180° + α
3. IV = 360° - α
Contoh Soal :
1. Nyatakan A (-4 , 2√2 ) dalam koordinat kutub
Jawaban :
r = √{(-4)2 + (2√2)2}
= √(16 + 8)
= √24
= 2√6
Tan α = 2√2/-4
= -1/2 √2 (kuadran II)
= 180° - 35°
= 145°
Jadi, A (2√6 , 145°)
Tan α = -2/-2
α = 1 (kuadran III)
α = 180° + 45°
α = 225°
Cos α = x/r
2. Nyatakan B (-2 , -2) dalam koordinat kutub
Jawaban : r = √{(-2)2 + (-2)2}
= √(4 + 4)
= √8
= 2 √2Tan α = -2/-2
α = 1 (kuadran III)
α = 180° + 45°
α = 225°
Jadi, B (2√2 , 225° )
* Mengubah Koordinat Kutub ke Kartesius *
Cos α = x/r x = r cos α
Sin α = y/r
y = r sin α
Contoh Soal :
1. Nyatakan C (10 , 240) dalam koordinat kutub
Jawaban :
x = r cos α dan y = r sin α
= 10 cos 240° = 10 sin 240°
= 10 (-1/2) = 10 (-1/2 √3)
= -5 = -5 √3
Jadi, C (-5 , -5 √3)
Catatan : Ciri utama aturan Sinus yang diketahui harus ada sudut dan sisi di depan sudut, serta salah satu sudut. Maka sisi di depan sudut tersebut bisa dihitung.
* ATURAN SINUS *
a/sinα = b/sinβ = c/sinγ
Catatan : Ciri utama aturan Sinus yang diketahui harus ada sudut dan sisi di depan sudut, serta salah satu sudut. Maka sisi di depan sudut tersebut bisa dihitung.
30 Maret 2013 pukul 22.04
keren gan :D