Himpunan Penyelesaian Dari Tan^2 3x – 1 = 0

Himpunan penyelesaian dari tan² 3x – 1 = 0 untuk 0 < x < 2π adalah …

PEMBAHASAN

tan² 3x = 1

tan 3x = √1

tan 3x =  1 dan tan 3x = -1

tan 3x = tan π/4 dan tan 3x = tan 3π/4

 

tan 3x = tan π/4

3x = π/4 + k . π

x = π/12 + k . π/3 = π/12 + k . 4π/12

untuk

k = 0, maka x = π/12 + 0 . 4π/12 = π/12

k = 1, maka x = π/12 + 1 . 4π/12 = 5π/12

k = 2, maka x = π/12 + 2 . 4π/12 = 9π/12 = 3π/4

k = 3, maka x = π/12 + 3 . 4π/12 = 13π/12

k = 4, maka x = π/12 + 4 . 4π/12 = 17π/12

k = 5, maka x = π/12 + 5 . 4π/12 = 21π/12 = 7π/4

k = 6, maka x = π/12 + 6 . 4π/12 = 25π/12 (tidak memenuhi)

 

tan 3x = tan 3π/4

3x = 3π/4 + k . π

x = 3π/12 + k . π/3 = 3π/12 + k . 4π/12

k = 0, maka x = 3π/12 + 0 . 4π/12 = 3π/12 = π/4

k = 1, maka x = 3π/12 + 1 . 4π/12 = 7π/12

k = 2, maka x = 3π/12 + 2 . 4π/12 = 11π/12

k = 3, maka x = 3π/12 + 3 . 4π/12 = 15π/12 = 5π/4

k = 4, maka x = 3π/12 + 4 . 4π/12 = 19π/12

k = 5, maka x = 3π/12 + 5 . 4π/12 = 23π/12

k = 6, maka x = 3π/12 + 6 . 4π/12 = 27π/12 (tidak memenuhi)

 

HP = { π/12 , π/4 , 5π/12 , 7π/12 , 3π/4 , 11π/12 , 13π/12 , 5π/4 , 17π/12 , 19π/12 , 7π/4 

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