The general solution of the system of equations sin 3 x + sin 3 2 π 3 + x + sin 3 4 π 3 + x + 3 4 cos 2 x = 0 and cos ⁡ x ≠ 0 is

Question

The general solution of the system of equations sin 3 x + sin 3 2 π 3 + x + sin 3 4 π 3 + x + 3 4 cos 2 x = 0 and cos ⁡ x ≠ 0 is, x = 2 k + 1 π 10 , k ∈ Z, x = 2 k + 1 π 5 , k ∈ Z, x = 4 k + 1 π 10 , k ∈ Z, x = 4 k + 1 5 π , k ∈ Z

MARKS App · Accepted Answer

∵ sin 3 x = 3 sinx - 4 sin 3 x ⇒ sin 3 x = 1 4 3 sinx - sin 3 x The given equation reduces to 1 4 3 sinx - sin 3 x + 1 4 3 sin 2 π 3 + x - sin 2 π + 3 x + 1 4 ( 3 sin 4 π 3 + x - sin 4 π + 3 x + 3 4 cos 2 x = 0 3 sin ⁡ x + sin 2 π 3 + x + sin ⁡ 4 π 3 + x - sin ⁡ 3 x + sin ⁡ 2 π + 3 x + sin ⁡ 4 π + 3 x + 3 cos ⁡ 2 x = 0 3 sinx + 2 sinx cos ⁡ 2 π 3 - 3 sin 3 x + 3 cos 2 x = 0 ⇒ sin 3 x = cos 2 x ⇒ cos π 2 - 3 x = cos 2 x ⇒ 2 x = 2 kπ ± π 2 - 3 x ⇒ 5 x = 2 k π + π 2 = 4 k + 1 π 2 ∀ k ∈ Z or x =-2 kπ + π 2 , ∀ k ∈ Z ⇒ x = 4 k + 1 π 10 , ∀ k ∈ Z cos ⁡ x ≠ 0

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