Question

Решите уравнение
1+sin 2x = 2 sin x + cos x.
Срочно дам 20 баллов​

Answer 1

1 + Sin2x = 2Sinx + Cosx

1 + 2SinxCosx - 2Sinx - Cosx = 0

(2SinxCosx - 2Sinx) + ( 1 - Cosx) = 0

2Sinx(Cosx - 1) + (1 - Cosx) = 0

2Sinx(Cosx - 1) - (Cosx - 1) = 0

(Cosx - 1)(1Sinx - 1) = 0

$\left[\begin{array}{ccc}Cosx-1=0\\2Sinx-1=0\end{array}\right\\\\\\\left[\begin{array}{ccc}Cosx=1\\Sinx=\frac{1}{2} \end{array}\right\\\\\\\left[\begin{array}{ccc}x=2\pi n,n\in Z \\x=(-1)^{n}arcSin\frac{1}{2}+\pi n,n\in Z\end{array}\right\\\\\\\left[\begin{array}{ccc}x=2\pi n,n\in Z \\x=(-1)^{n}\frac{\pi }{6}+\pi n,n\in Z \end{array}\right$