Question

Решите уравнение разложением на множители: 1)cos(2(x + 60°)) + 4sin(x + 60°) = 2,5;​

Answer 1

$\displaystyle\bf\\Cos\Big(2(x+60^\circ)\Big)+4Sin(x+60^\circ)=2,5\\\\1-2Sin^{2} (x+60^\circ)+4Sin(x+60^\circ)=2,5\\\\-2Sin^{2} (x+60^\circ)+4Sin(x+60^\circ)-1,5=0 \ |\cdot(-2)\\\\4Sin^{2} (x+60^\circ)-8Sin(x+60^\circ)+3=0\\\\4\Big(Sin^{2} (x+60^\circ)-2Sin(x+60^\circ+1\Big)-4+3=0\\\\4\Big(Sin(x+60^\circ)-1\Big)^{2} -1=0$

$\displaystyle\bf\\\Big(2Sin(x+60^\circ)-2-1\Big)\cdot\Big(2Sin(x+60^\circ)-2+1\Big)=0\\\\1)\\\\2Sin(x+60^\circ)-2-1=0\\\\2Sin(x+60^\circ)=3\\\\Sin\Big(x+60^\circ\Big)=1,5\\\\x\in\oslash\\\\2)\\\\2Sin(x+60^\circ)-2+1=0\\\\Sin(x+60^\circ)=\frac{1}{2} \\\\x+60^\circ=(-1)^{n} 30^\circ+180^\circ n\\\\x=(-1)^{n} 30^\circ -60^\circ+180^\circ n$