х₁ = πn; x₂ = π/6 + 2πk; x₃ = 5π/6 + 2πm; n, k, m ∈ Z
Корни - 19π/6; -3π; -2π;
Объяснение:
sinx + 2sin(2x + π/6) = √3sin2x + 1
sinx + 2(sin2x · cos π/6 + cos2x · sin π/6) = √3sin2x + 1
sinx + 2(sin2x · 0.5√3 + cos2x ·0.5) = √3sin2x + 1
sinx + √3sin2x + cos2x = √3sin2x + 1
sinx + cos2x = 1
sinx + 1 - 2sin²x = 1
2sin²x - sinx = 0
sinx (2sinx - 1) = 0
1) sin x = 0
x = πn
б) корни на интервале [-7π/2; -2π]
--7π/2 ≤ πn ≤ -2π
-3.5 ≤ n ≤ -2
n = -3 и n = -2
корни -3π и -2π
2) 2sinx - 1 = 0
sinx = 1/2
x₂ = π/6 + 2πk
x₃ = 5π/6 + 2πm
б) корни на интервале [-7π/2; -2π]
-7π/2 ≤ π/6 + 2πk ≤ -2π
-7/2 - 1/6 ≤ 2k ≤ -2 - 1/6
-22/6 ≤ 2k ≤ -13/6
-11/6 ≤ k ≤ -13/12
≤ k ≤ 
Корней нет
-7π/2 ≤ 5π/6 + 2πm ≤ -2π
-7/2 - 5/6 ≤ 2m ≤ -2 - 5/6
-26/6 ≤ 2m ≤ -17/6
-13/6 ≤ m ≤ -17/12
≤ m ≤ 
m = -2
Корень
5π/6 - 4π = - 19π/6
