Question

Tập giá trị T của hàm số \(y=\sin 2019x-\cos 2019x  \)

\(A. T=\left[-2;2\right]. B. T=\left[-4034;4034\right].\)

\(C. T=\left[-\sqrt{2} ;\sqrt{2} \right]. D. T=\left[0;\sqrt{2} \right].\)
 

Answer 1

Chọn C
\(y=\sin 2019x-\cos 2019x=\sqrt{2} \sin \left(2019x-\frac{\pi }{4} \right). \)

 Ta có: 
\(\begin{array}{l} {-1\le \sin \left(2019x-\frac{\pi }{4} \right)\le 1\quad \forall x} \\ {\Leftrightarrow -\sqrt{2} \le \sqrt{2} \sin \left(2019x-\frac{\pi }{4} \right)\le \sqrt{2} \quad \forall x\Leftrightarrow -\sqrt{2} \le y\le \sqrt{2} \quad \forall x.} \\ {y=-\sqrt{2} \Leftrightarrow \sin \left(2019x-\frac{\pi }{4} \right)=-1\Leftrightarrow x=-\frac{\pi }{8076} +\frac{2k\pi }{2019} \quad \left(k\in {\rm Z}\right).} \end{array} \)
\(y=\sqrt{2} \Leftrightarrow \sin \left(2019x-\frac{\pi }{4} \right)=1\Leftrightarrow x=\frac{3\pi }{8076} +\frac{2k\pi }{2019} \quad \left(k\in {\rm Z}\right). \)

 Vậy \(T=\left[-\sqrt{2} ;\sqrt{2} \right].\)