Toán Lớp 11: tính tổng T tất cả các nghiệm của phương trình 2sin^2x/4-3cosx/4=0 trên [0;8pi]
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Toán Lớp 11: tính tổng T tất cả các nghiệm của phương trình 2sin^2x/4-3cosx/4=0 trên [0;8pi]
Comments ( 1 )
\quad 2\sin^2\dfrac{x}{4} – 3\cos\dfrac{x}{4} = 0\\
\Leftrightarrow 2\cos^2\dfrac{x}{4} + 3\cos\dfrac{x}{4} – 2 =0\\
\Leftrightarrow \left(2\cos\dfrac{x}{4} – 1\right)\left(\cos\dfrac{x}{4} + 2\right) = 0\\
\Leftrightarrow \left[\begin{array}{l}\cos\dfrac{x}{4} = \dfrac12\\\cos\dfrac{x}{4} = -2\quad (vn)\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}\dfrac{x}{4} = \dfrac{\pi}{3} + k2\pi\\\dfrac{x}{4} = – \dfrac{\pi}{3} + k2\pi\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = \dfrac{4\pi}{3} + k8\pi\\x = – \dfrac{4\pi}{3} + k8\pi\end{array}\right.\quad (k\in\Bbb Z)\\
\text{Ta có:}\\
\quad 0 \leqslant x \leqslant 8\pi\\
\Leftrightarrow \left[\begin{array}{l}0 \leqslant \dfrac{4\pi}{3} + k8\pi \leqslant 8\pi\\0 \leqslant -\dfrac{4\pi}{3} + k8\pi \leqslant 8\pi\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}-\dfrac16 \leqslant k \leqslant \dfrac56\\\dfrac16 \leqslant k \leqslant \dfrac76\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}k = 0 \Rightarrow x = \dfrac{4\pi}{3}\\k = 1 \Rightarrow x = \dfrac{20\pi}{3}\end{array}\right.\\
\Rightarrow \dfrac{4\pi}{3} + \dfrac{20\pi}{3} = 8\pi
\end{array}\)