Trigonometri sin (x-pi/3) + sin (x-4pi/3) senilai dengan

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Pembahasan

Untuk menyederhanakan ekspresi trigonometri sin(x - π/3) + sin(x - 4π/3), kita dapat menggunakan rumus penjumlahan sinus:
sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2).

Dalam kasus ini, A = x - π/3 dan B = x - 4π/3.

1. Hitung (A+B)/2:
   (A+B)/2 = ((x - π/3) + (x - 4π/3)) / 2
   (A+B)/2 = (2x - 5π/3) / 2
   (A+B)/2 = x - 5π/6

2. Hitung (A-B)/2:
   (A-B)/2 = ((x - π/3) - (x - 4π/3)) / 2
   (A-B)/2 = (x - π/3 - x + 4π/3) / 2
   (A-B)/2 = (3π/3) / 2
   (A-B)/2 = π / 2

3. Substitusikan kembali ke rumus penjumlahan sinus:
sin(x - π/3) + sin(x - 4π/3) = 2 sin(x - 5π/6) cos(π/2)

Karena cos(π/2) = 0, maka:
sin(x - π/3) + sin(x - 4π/3) = 2 sin(x - 5π/6) * 0
sin(x - π/3) + sin(x - 4π/3) = 0

Jadi, sin(x - π/3) + sin(x - 4π/3) senilai dengan 0.