The value of θ for which the system of equations sin ⁡ 3 θ x - 2 y + 3 z = 0,   ( cos ⁡ 2 θ   x + 8 y - 7 z = 0 and 2 x + 14 y - 11 z = 0 has a non-trivial solution, is (here, n ∈ Z )

Step-by-step solution

The system of equations has a non-trivial solution if and only if sin ⁡ 3 θ - 2 3 cos ⁡ 2 θ 8 - 7 2 14 - 11 = 0 Applying R 2 → R 2 + 4 R 1 , R 3 → R 3 + 7 R 1 , we get, sin ⁡ 3 θ - 2 3 cos ⁡ 2 θ + 4 sin ⁡ 3 θ 0 5 2 + 7 sin ⁡ 3 θ 0 10 = 0 Expanding along C 2 , we get, 2 cos ⁡ 2 θ + 4 sin ⁡ 3 θ - 2 + 7 sin ⁡ 3 θ = 0 ⇒ 2 - 2 cos ⁡ 2 θ - sin ⁡ 3 θ = 0 ⇒ 4 sin 2 ⁡ θ - 3 sin ⁡ θ - 4 sin 3 ⁡ θ = 0 ⇒ sin ⁡ θ 4 sin 2 ⁡ θ + 4 sin ⁡ θ - 3 = 0 ⇒ sin ⁡ θ 2 sin ⁡ θ - 1 2 sin ⁡ θ + 3 = 0 ⇒ sin ⁡ θ = 0 or sin ⁡ θ = 1 / 2 . [ sin ⁡