TRIGONOMETRI
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 diketahuiharus 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 diketahuiharus ada sudut dan sisi di depan sudut, serta salah satu sudut. Maka sisi di depan sudut tersebut bisa dihitung.
