Уравнение sin4x*sin6x=2 (sinx+sin5x)

Решим уравнение, воспользовавшись формулами:

sin2a = 2sina * cosa; sina + sinb = 2sin ((a + b) / 2) * cos ((a - b) / 2); cosa + cosb = 2cos ((a + b) / 2) * cos ((a - b) / 2); sin4x * sin6x = 2 (sinx + sin5x); 2sin2x * cos2x * 2sin3x * cos3x = 4 * sinx2x * sin3x; sin2x * cos2x * sin3x * cos3x - sinx2x * sin3x = 0; sin2x * sin3x (cos2x * cos3x - 1) = 0; sin2x * sin3x (2cos2x * cos3x - 2) = 0; sin2x * sin3x (cosx + cos5x - 2) = 0; [sin2x * sin3x = 0;

[cosx + cos5x - 2 = 0; [sin2x = 0;

[sin3x = 0;

[cosx + cos5x = 2; [sin2x = 0;

[sin3x = 0;

[cosx = 1;

[cos5x = 1; [2x = πk;

[3x = πk;

[x = 2πk;

[5x = 2πk; [x = πk/2;

[x = πk/3;

[x = 2πk/5.

Ответ: πk/2; πk/3; 2πk/5, k ∈ Z.