Question

Решите уравнение:
3sin^3x + 4sin^2x cos x - sinx cos^2x = 2sinx + 3cos x

Answer 1

$3sin^3x + 4sin^2x cos x - sin x cos^2x = 2sin x + 3cos x;\3sin^3x + 4sin^2x cos x - sin x cos^2x = 2(sin^2x+cos^2x)sin x + 3(sin^2x+cos^2x)cos x;\sin^3x + sin^2x cos x - 3sin x cos^2x -3cos^3x=0;\sin^2x(sin x+cos x)-3cos^2x(sin x+cos x)=0;\(1-cos^2 x)(sin x+cos x)-3cos^2x(sin x+cos x)=0;\(1-4cos^2 x)(sin x+cos x)=0;$

$left[begin{matrix}cos x=pm frac{1}{2},\ sin (x+frac{pi}{4})=0;end{matrix}right.Rightarrow left[begin{matrix} x=pm frac{pi}{3}+2pi n,,nin Z,\x=pm frac{2pi}{3}+2pi m,,min Z,\ x=-frac{pi}{4}+pi k,,kin Z.end{matrix}right.Rightarrow$