Sin3x+sin2x+sinx=0

sinx+sin3x+2cosx=0

cos9x-cos7x+cos3x-cosx=0

Решить тригонометрические уравнения

1) Sin3x+Sin2x+Sinx=0

2Sim2xCosx + Sin2x = 0

Sin2x (Cosx + 1) = 0

Sin2x = 0 или Cosx + 1 = 0

2x = πn, n ∈Z Cosx = - 1

x = πn/2, n ∈Z x = π + 2πk, k ∈ Z

2) Sinx+Sin3x+2cosx=0

2Sim2xCosx + 2Cosx = 0

Cosx (2Sin2x + 2) = 0

Cosx = 0 или 2Sin2x + 2 = 0

x = π/2 + πk, k ∈Z Sinx = - 1

x = - π/2 + nπ, n ∈Z

3) Cos9x-Cos7x+Cos3x-Cosx=0

-2Sin5xSin4x - 2Sin5xSin2x = 0

Sin5xSin4x + Sin5xSin2x = 0

Sin5x (Sin4x + Sin2x) = 0

Sin5x = 0 или Sin4x + Sin2x = 0

5x = πn, n ∈Z 2Sin3xCosx = 0

x = πn/5, n ∈Z Sin3x = 0 или Cosx = 0

3x = πk, k ∈Z x = π/2 + πm, m∈Z

x = πk/3, k ∈Z

ответ: x = π/2 + πm, m∈Z (все остальные сюда входят)