Question

Tentukan himpunan penyelesaian dari setiap soal berikut :
Persamaan trigonometri sederhana

Answer 1

Verified answer

Persamaan Trigonometri

[10] 3 tan x + 3 = 0 dalam interval 0 ≤ x ≤ π
tan x = -1
tan x = tan (π - π/4)
tan x = tan 3π/4
x = 3π/4 + k.π
k = 0 ⇒ x = 3π/4
k = 1 ⇒ x = 7π/4 
∴ HP = {3π/4, 7π/4}

[11] sin 2x = ¹/₂ dalam interval 0° ≤ x ≤ 360°
sin 2x = sin 30°
Bagian-1
2x = 30° + k.360°
x = 15° + k.180°
k = 0 ⇒ x = 15°
k = 1 ⇒ x = 195°
k = 2 ⇒ x = 375° tidak memenuhi karena di luar interval
Bagian-2
2x = (180°-30°) + k.360°
x = 75° + k.180°
k = 0 ⇒ x = 75° 
k = 1 ⇒ x = 255° 
k = 2 ⇒ x = 435° tidak memenuhi karena di luar interval
∴ HP = {15°, 75°, 195°, 255°}

[12] cos 3x = ¹/₂ dalam interval 0 ≤ x ≤ 2π
cos 3x = cos π/3
Bagian-1
3x = π/3 + k.2π
x = π/9 + k.(²/₃)π
k = 0 ⇒ x = π/9
k = 1 ⇒ x = 7π/9
k = 2 ⇒ x = 13π/9 
Bagian-2
3x = -π/3 + k.2π
x = -π/9 + k.(²/₃)π
k = 1 ⇒ x = 5π/9
k = 2 ⇒ x = 11π/9 
k = 3 ⇒ x = 17π/9
∴ HP = {π/9, 5π/9, 7π/9, 11π/9, 13π/9, 17π/9}

[13] 3 tan 2x = √3 dalam interval 0 ≤ x ≤ π
tan 2x = ¹/₃.√3
tan 2x = tan π/6
2x = π/6 + k.π
x = π/12 + kπ/2
k = 0 ⇒ x = π/12
k = 1 ⇒ x = 7π/12 
∴ HP = {π/12, 7π/12}

[14] sin (x + 30°) = ¹/₂ dalam interval 0° ≤ x ≤ 360°
sin (x + 30) = sin 30°
Bagian-1
x + 30° = 30° + k.360°
x = k.360°
k = 0°
k = 1 ⇒ x = 360°
Bagian-2
x + 30° = (180°- 30°) + k.360°
x = 120° + k.360°
k = 0 ⇒ x = 120° 
∴ HP = {0°, 120°, 360°}