Cho hàm số \(y=\sqrt{\left( 2m-1 \right)\sin x-\left( m+2 \right)\cos x+4m-3}\,\)(1).

Hàm số (1) xác định \(\forall x\in \mathbb{R}\)\(\Leftrightarrow \left( 2m-1 \right)\sin x-\left( m+2 \right)\cos x+4m-3\ge 0,\forall x\) (2)

+ Nếu \(x=\pi +k2\pi \), (2) \(\Leftrightarrow m+2+4m-3\ge 0\)\(\Leftrightarrow m\ge \frac{1}{5}\)

+ Nếu \(x\ne \pi +k2\pi \), đặt \(t=\tan \frac{x}{2}\). Ta có: \(\sin x=\frac{2t}{1+{{t}^{2}}}\), \(\cos x=\frac{1-{{t}^{2}}}{1+{{t}^{2}}}\)

(2) \(\Leftrightarrow \left( 2m-1 \right).\frac{2t}{1+{{t}^{2}}}-\left( m+2 \right)\frac{1-{{t}^{2}}}{1+{{t}^{2}}}+4m-3\ge 0,\forall t\)

\(\Leftrightarrow \left( 5m-1 \right){{t}^{2}}+2\left( 2m-1 \right)t+3m-5\ge 0,\,\forall t\) (*)

TH1: \(m=\frac{1}{5}\), (*)\(\Leftrightarrow -\frac{6}{5}t-\frac{22}{5}\ge 0\)\(\Leftrightarrow t\le -\frac{11}{3}\) (KTM – loại)

TH2: \(m\ne \frac{1}{5}\)

(*) \(\begin{array}{l}
 \Leftrightarrow \left\{ \begin{array}{l}
m > \frac{1}{5}\\
\Delta ' = {\left( {2m - 1} \right)^2} - \left( {5m - 1} \right)\left( {3m - 5} \right) \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > \frac{1}{5}\\
 - 11{m^2} + 24m - 4 \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > \frac{1}{5}\\
\left[ \begin{array}{l}
m \le \frac{2}{{11}}\\
m \ge 2
\end{array} \right.
\end{array} \right.
\end{array}\)

Do đó \(\left( 2 \right)\Leftrightarrow m\ge 2\).

Lại có: \(\left\{ \begin{array}{l}
m > 0\\
m < 2019\\
m \in Z*
\end{array} \right.\) nên \(m \in \left\{ {2{\mkern 1mu} ,{\mkern 1mu} 3{\mkern 1mu} ,{\mkern 1mu} 4{\mkern 1mu} ,...,2018} \right\}\)

Vậy có 2017 giá trị m cần tìm.