Решить уравнение: cos3x+cos2x=sin5x

Решение

cos3x+cos2x=sin5x

2cos (3x + 2x) / 2 * cos (3x - 2x) / 2 = sin5x

2cos2,5x * cos0,5x - sin2*2,5x = 0

2*cos2,5x * cos0,5x - 2*sin2,5x * cos2,5x = 0

2cos2,5x * (cos0,5x - sin2,5x) = 0

1) cos2,5x = 0

2,5x = π + πk, k ∈ Z

x₁ = 2π/5 + 2πk/5, k ∈ Z

2) cos0,5x - sin2,5x = 0

cos0,5x - cos (π/2 - 2,5x) = 0

- 2sin (0,5x + 2,5x) / 2 * sin (0,5x - 2,5x) / 2 = 0

3) sin1,5x = 0

1,5x = πn, n ∈ Z

x₂ = 2πn/3, n ∈ Z

4) sinx = 0

x₃ = πm, m ∈ Z