Question

Решите уравнение sin x sin 3x = 1/2

Answer 1

Решите уравнение sin x sin 3x = 1/2

Answer 2

$sinx*sin3x=dfrac{1}{2}\\Formyla;:\\boxed{;;sinalpha*sinbeta=dfrac{1}{2}*Big(;cosbig(alpha-betabig)-cosbig(alpha+betabig);Big);;}\\\boxed{big{;;cos2x=2cos^2x-1;;}}\\\dfrac{1}{2}*Big(cos2x-cos4xBig)=dfrac{1}{2}\\cos2x-cos4x=1\\cos2x-Big(2cos^22x-1Big)=1\\cos2x-2cos^22x+1=1\\2cos^22x-cos2x=0\\cos2x*Big(;2cos2x-1;Big)=0\\cos2x*Big(;cos2x-dfrac{1}{2};Big)=0$

$1);;;cos2x=0\\2x=dfrac{pi}{2}+pi;n;;,;;;n;in;mathbb{Z}\\boxed{;;x;_k=dfrac{pi}{4}+frac{pi;k}{2};;,;;;n;in;mathbb{Z};;}\\\2);;;cos2x-dfrac{1}{2}=0\\cos2x=dfrac{1}{2}\\2x=pm;dfrac{pi}{3}+2pi;p;;,;;;p;in;mathbb{Z}\\boxed{Big{;;x;_m=pm;dfrac{pi}{6}+pi;m;;,;;;m;in;mathbb{Z};;}}$