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प्रश्न
The possible values of θ ∈ (0, π) such that sin (θ) + sin (4θ) + sin (7θ) = 0 are ______.
विकल्प
`π/4, (5π)/12, π/2, (2π)/3, (3π)/4, (8π)/9`
`(2π)/9, π/4, π/2, (2π)/3, (3π)/4, (35π)/36`
`(2π)/9, π/4, π/2, (2π)/3, (3π)/4, (8π)/9`
`(2π)/9, π/4, (4π)/9, π/2, (3π)/4, (8π)/9`
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उत्तर
The possible values of θ ∈ (0, π) such that sin (θ) + sin (4θ) + sin (7θ) = 0 are `underlinebb((2π)/9, π/4, (4π)/9, π/2, (3π)/4, (8π)/9)`.
Explanation:
sin(θ) + sin(4θ) + sin(7θ) = 0
⇒ sin(4θ − 4θ) + sin(4θ) + sin(4θ + 3θ) = 0
⇒ 2 sin 4θ cos 3θ + sin 4θ = 0
⇒ sin 4θ(2 cos 3θ + 1) = 0
⇒ sin 4θ = 0 or cos 3θ = `(-1)/2`
θ ∈ (0, π) ⇒ 4θ ∈ (0, 4π)
∴ 4θ = π, 2π, 3π where sin 4θ = 0
∴ θ = `π/4, (2π)/4, (3π)/4`
θ ∈ (0, π) ⇒ 3θ ∈ (0, 3π)
∴ 3θ = `(2π)/3, (4π)/3, (8π)/3` where cos 3θ = `(-1)/2`
θ = `(2π)/9, (4π)/9, (8π)/9`
∴ Possible values of θ are `π/4, π/2, (3π)/4, (2π)/9, (4π)/9, (8π)/9`
