(1)РЕШЕНИЕ:
![sin(2x)+sin(x)=2cos(x)+1; \ [\pi;3\pi]\\ sin(x)\cdot (2cos(x)+1)=2cos(x)+1;\\ (2cos(x)+1) \cdot (sin(x)-1) = 0;\\ \begin{cases} cos(x)=-\frac{1}{2}\\sin(x)=1 \end{cases}\\ \begin{cases} x=\left[\begin{array}{ccc}arccos(-\frac{1}{2})+2\pi k \\ -arccos(-\frac{1}{2}) +2\pi n \end{array}\right\\x=\left[\begin{array}{ccc}arcsin(1)+2\pi l\\ \pi - arcsin(1)+2\pi m\end{array}\right\end{cases}\\](https://tex.z-dn.net/?f=sin%282x%29%2Bsin%28x%29%3D2cos%28x%29%2B1%3B+%5C+%5B%5Cpi%3B3%5Cpi%5D%5C%5C+sin%28x%29%5Ccdot+%282cos%28x%29%2B1%29%3D2cos%28x%29%2B1%3B%5C%5C+%282cos%28x%29%2B1%29+%5Ccdot+%28sin%28x%29-1%29+%3D+0%3B%5C%5C+%5Cbegin%7Bcases%7D+cos%28x%29%3D-%5Cfrac%7B1%7D%7B2%7D%5C%5Csin%28x%29%3D1+%5Cend%7Bcases%7D%5C%5C+%5Cbegin%7Bcases%7D+x%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Darccos%28-%5Cfrac%7B1%7D%7B2%7D%29%2B2%5Cpi+k+%5C%5C+-arccos%28-%5Cfrac%7B1%7D%7B2%7D%29+%2B2%5Cpi+n+%5Cend%7Barray%7D%5Cright%5C%5Cx%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Darcsin%281%29%2B2%5Cpi+l%5C%5C+%5Cpi+-+arcsin%281%29%2B2%5Cpi+m%5Cend%7Barray%7D%5Cright%5Cend%7Bcases%7D%5C%5C "sin(2x)+sin(x)=2cos(x)+1; \ [\pi;3\pi]\\ sin(x)\cdot (2cos(x)+1)=2cos(x)+1;\\ (2cos(x)+1) \cdot (sin(x)-1) = 0;\\ \begin{cases} cos(x)=-\frac{1}{2}\\sin(x)=1 \end{cases}\\ \begin{cases} x=\left[\begin{array}{ccc}arccos(-\frac{1}{2})+2\pi k \\ -arccos(-\frac{1}{2}) +2\pi n \end{array}\right\\x=\left[\begin{array}{ccc}arcsin(1)+2\pi l\\ \pi - arcsin(1)+2\pi m\end{array}\right\end{cases}\\")
\begin{cases} x=\left[\begin{array}{ccc}\frac{2\pi}{3}+2\pi k \\ -\frac{2\pi}{3} +2\pi n \end{array}\right\\ x = \frac{\pi}{2}+2\pi m\end{cases} " alt="\begin{cases} x=\left[\begin{array}{ccc}\frac{2\pi}{3}+2\pi k \\ -\frac{2\pi}{3} +2\pi n \end{array}\right\\x=\left[\begin{array}{ccc}\frac{\pi}{2}+2\pi l\\ \frac{\pi}{2}+2\pi m\end{array}\right\end{cases} <=> \begin{cases} x=\left[\begin{array}{ccc}\frac{2\pi}{3}+2\pi k \\ -\frac{2\pi}{3} +2\pi n \end{array}\right\\ x = \frac{\pi}{2}+2\pi m\end{cases} " align="absmiddle" class="latex-formula">
(2)ВЫБОРКА:
k = 1; \ x = \frac{8\pi}{3}\\ \\ 2) \ \pi \leq -\frac{2\pi}{3}+2\pi n \leq 3\pi;\ \ \frac{5}{6}\leq n \leq \frac{11}{6}\\ => n = 1; \ x = \frac{4\pi}{3}\\ \\ 3) \ \pi \leq \frac{\pi}{2}+2\pi m \leq 3\pi;\ \ \frac{1}{4}\leq m \leq \frac{5}{4}\\ => m = 1; \ x = \frac{5\pi}{2}\\ \\ " alt="1) \ \pi \leq \frac{2\pi}{3}+2\pi k \leq 3\pi;\ \ \frac{1}{6}\leq k \leq \frac{7}{6}\\ => k = 1; \ x = \frac{8\pi}{3}\\ \\ 2) \ \pi \leq -\frac{2\pi}{3}+2\pi n \leq 3\pi;\ \ \frac{5}{6}\leq n \leq \frac{11}{6}\\ => n = 1; \ x = \frac{4\pi}{3}\\ \\ 3) \ \pi \leq \frac{\pi}{2}+2\pi m \leq 3\pi;\ \ \frac{1}{4}\leq m \leq \frac{5}{4}\\ => m = 1; \ x = \frac{5\pi}{2}\\ \\ " align="absmiddle" class="latex-formula">
(3)ОТВЕТ: 