3sin2х - 4cоsх + 3sinх - 2 = 0
6sinх·cоsх - 4cоsх + 3sinх - 2 = 0
(6sinх·cоsх + 3sinх) - (4cоsх + 2) = 0
3sinх·(2cоsх + 1) - 2·(2cоsх + 1) = 0
(2cоsх + 1)·(3sinх - 2) = 0
1) 2cоsх + 1 = 0
cоsх = -1/2
x₁ = 4π/3 + 2πn
x₂ = -4π/3 + 2πn
2) 3sinх - 2 = 0
sinх = 2/3
x₃ = (-1)^k ·arcsin(2/3) + πk
Исследуем х₁ = 4π/3 + 2πn
n = 0 x₁ = 4π/3 x∈[π/2; 3π/2]
n = 1 x₁ = 4π/3 + 2π x∉[π/2; 3π/2]
Исследуем x₂ = -4π/3 + 2πn
n = 1 x₂ = -4π/3 + 2π = 2π/3 x∈[π/2; 3π/2]
n = 2 x₂ = -4π/3 + 4π = 8π/3 x∉[π/2; 3π/2]
Исследуем x₃ = (-1)^k ·arcsin(2/3) + πk
arcsin(2/3) ≈ 42°
n = 1 x₃ = -arcsin(2/3) + π ≈ 138° x∈[π/2; 3π/2]
n = 2 x₃ = arcsin(2/3) + 2π ≈ 402° x∉[π/2; 3π/2]
ответ: в интервале x∈[π/2; 3π/2] уравнеие имеет три корня
x₁ = 4π/3, x₂ = 2π/3, x₃ = -arcsin(2/3) + π
3sin2х - 4cоsх + 3sinх - 2 = 0
6sinх·cоsх - 4cоsх + 3sinх - 2 = 0
(6sinх·cоsх + 3sinх) - (4cоsх + 2) = 0
3sinх·(2cоsх + 1) - 2·(2cоsх + 1) = 0
(2cоsх + 1)·(3sinх - 2) = 0
1) 2cоsх + 1 = 0
cоsх = -1/2
x₁ = 4π/3 + 2πn
x₂ = -4π/3 + 2πn
2) 3sinх - 2 = 0
sinх = 2/3
x₃ = (-1)^k ·arcsin(2/3) + πk
Исследуем х₁ = 4π/3 + 2πn
n = 0 x₁ = 4π/3 x∈[π/2; 3π/2]
n = 1 x₁ = 4π/3 + 2π x∉[π/2; 3π/2]
Исследуем x₂ = -4π/3 + 2πn
n = 1 x₂ = -4π/3 + 2π = 2π/3 x∈[π/2; 3π/2]
n = 2 x₂ = -4π/3 + 4π = 8π/3 x∉[π/2; 3π/2]
Исследуем x₃ = (-1)^k ·arcsin(2/3) + πk
arcsin(2/3) ≈ 42°
n = 1 x₃ = -arcsin(2/3) + π ≈ 138° x∈[π/2; 3π/2]
n = 2 x₃ = arcsin(2/3) + 2π ≈ 402° x∉[π/2; 3π/2]
ответ: в интервале x∈[π/2; 3π/2] уравнеие имеет три корня
x₁ = 4π/3, x₂ = 2π/3, x₃ = -arcsin(2/3) + π