Himpunan Penyelesaian Dari 4 Sin^2 2x – 3 = 0

Himpunan penyelesaian dari 4 sin2 2x – 3 = 0 untuk 0 < x < 2π adalah …

PEMBAHASAN

4 sin2 2x = 3

sin2 2x = ¾

sin2 2x = √ ¾

sin 2x = ½ √3 dan sin 2x = - ½ √3

sin 2x = sin π/3 dan sin 2x = sin 4π/3

sin 2x = sin π/3

2x1 = π/3 + k . 2π

x1 = π/6 + k . π = π/6 + k . 6π/6

untuk:

k = 0, maka x1 = π/6 + 0 . 6π/6 = π/6

k = 1, maka x1 = π/6 + 1 . 6π/6 = 7π/6

k = 2, maka x1 = π/6 + 2 . 6π/6 = 13π/6 (tidak memenuhi)

2x2 = π - π/3 + k . 2π

x2 = 2π/6 + k . π = 2π/6 + k . 6π/6

untuk:

k = 0, maka x2 = 2π/6 + 0 . 6π/6 = 2π/6 = π/3

k = 1, maka x2 = 2π/6 + 1 . 6π/6 = 8π/6 = 4π/3

k = 2, maka x2 = 2π/6 + 2 . 6π/6 = 14π/6 (tidak memenuhi)

sin 2x = sin 4π/3

2x1 = 4π/3 + k . 2π

x1 = 4π/6 + k . π = 2π/3 + k . 3π/3

untuk:

k = 0, maka x1 = 2π/3 + 0 . 3π/3 = 2π/3

k = 1, maka x1 = 2π/3 + 1 . 3π/3 = 5π/3

k = 2, maka x1 = 2π/3 + 2 . 3π/3 = 8π/3 (tidak memenuhi)

2x2 = π - 4π/3 + k . 2π

x2 = -π/6 + k . π = -π/6 + k . 6π/6

untuk:

k = 0, maka x2 = -π/6 + 0 . 6π/6 = -π/6 (tidak memenuhi)

k = 1, maka x2 = -π/6 + 1 . 6π/6 = 5π/6

k = 2, maka x2 = -π/6 + 2 . 6π/6 = 11π/6

HP = { π/6 , π/3 , 2π/3 , 5π/6 , 7π/6 , 4π/3 , 5π/3 , 11π/6}