68.
\(0< x< \dfrac{\pi}{2}\Rightarrow\dfrac{\pi}{4}< x+\dfrac{\pi}{4}< \dfrac{3\pi}{4}\)
\(\Rightarrow\dfrac{\sqrt{2}}{2}< sin\left(x+\dfrac{\pi}{4}\right)\le1\)
\(\Rightarrow1< \sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)\le\sqrt{2}\)
\(\Rightarrow1< m\le\sqrt{2}\)
69.
\(\Leftrightarrow\dfrac{1}{2}sin8x-\dfrac{1}{2}sin2x=\dfrac{1}{2}sin12x-\dfrac{1}{2}sin2x\)
\(\Leftrightarrow sin12x=sin8x\)
\(\Leftrightarrow\left[{}\begin{matrix}12x=8x+k2\pi\\12x=\pi-8x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{2}\\x=\dfrac{\pi}{20}+\dfrac{k\pi}{10}\end{matrix}\right.\)
\(\Rightarrow x=\left\{0;\dfrac{\pi}{2};\dfrac{\pi}{20};\dfrac{3\pi}{20};\dfrac{\pi}{4};\dfrac{7\pi}{20};\dfrac{9\pi}{20}\right\}\)
70.
\(sinx+sin3x+sin2x=0\)
\(\Leftrightarrow2sin2x.cosx+sin2x=0\)
\(\Leftrightarrow sin2x\left(2cosx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\2cosx+1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{2}\\x=\dfrac{2\pi}{3}+k2\pi\\x=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Rightarrow x=\left\{\dfrac{\pi}{2};\dfrac{2\pi}{3};\right\}\)
