ans

024739
$\{x|x\ne \dfrac{\pi}{6}+k\pi, \ k\in \mathbf{Z}\}$

024740
$[-1,\dfrac{5}{4}]$

024742
充分非必要

024741
$\dfrac{5\pi}{6}$

024743
$0$

024744
$[\dfrac{13\pi}{6},+\infty)$

011448
(1) $[0,\dfrac{\pi}{3}]$; (2) $\dfrac{24}{25}$

004308
(1) $AN=\dfrac{4\sqrt{3}}{3}\sin\theta$, $AM=\dfrac{4\sqrt{3}}{3}\sin(\theta+\dfrac{\pi}{3})$, $\theta\in (0,\dfrac{2\pi}{3})$; (2) 使$AM=AN=2\text{km}$, 才能使工厂产生的噪声对居民的影响最小

018453
(1) 最大值是$6$, 当且仅当$x=\pi+2k\pi$, $k\in \mathbf{Z}$时取到; 最小值是$-2$, 当且仅当$x=2k\pi$, $k\in \mathbf{Z}$时取到;\\
(2) 最大值是$1$, 当且仅大哥$x=0$时取到; 最小值是$-\dfrac{1}{2}$, 当且仅当$x=-\dfrac{4\pi}{3}$时取到

018454
最小正周期为$\pi$, 单调增区间是$[k\pi-\dfrac{\pi}{3}k\pi+\dfrac{\pi}{6}]$, $k\in \mathbf{Z}$

018456
(1) $\{x|\dfrac{\pi}{2}+2k\pi<x<\dfrac{3\pi}{2}+2k\pi, \ k\in \mathbf{Z}\}$; (2) $\{x|-\dfrac{3\pi}{4}+2k\pi<x<\dfrac{3\pi}{4}+2k\pi, \ k\in \mathbf{Z}\}$

009604
$\dfrac{1}{2}$

009605
(1) 奇函数, 理由略; (2) 奇函数, 理由略; (3) 既不是奇函数, 又不是偶函数, 理由略

009606
最小正周期是$4\pi$, 单调减区间为$[\dfrac{\pi}{3}+4k\pi,\dfrac{7\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$, 单调增区间为$[-\dfrac{5\pi}{3}+4k\pi,\dfrac{\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$

010289
(1) \begin{tikzpicture}[>=latex, scale = 0.6]
\draw [->] (-0.5,0) -- (7,0) node [below] {$x$};
\draw [->] (0,-3.5) -- (0,1.5) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw [domain = 0:2*pi, samples = 100] plot (\x,{cos(\x/pi*180)*2-1});
\draw [dashed] (2*pi,1) -- (0,1) node [left] {$1$};
\draw [dashed] (2*pi,1) -- (2*pi,0) node [below] {$2\pi$};
\draw [dashed] (pi,0) node [above] {$\pi$} -- (pi,-3) -- (0,-3) node [left] {$-3$};
\end{tikzpicture}\\
(2) \begin{tikzpicture}[>=latex]
\draw [->] (-7,0) -- (7,0) node [below] {$x$};
\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw [domain = -pi/2:pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
\draw [domain = pi/2:3*pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
\draw [domain = 3*pi/2:7, samples = 100] plot (\x,{cos(\x/pi*180)});
\draw [domain = -3*pi/2:-pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
\draw [domain = -7:-3*pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
\draw (pi/2,0) node [below] {$\frac{\pi}{2}$};
\draw (0,1) node [above left] {$1$};
\end{tikzpicture}

023606
$\dfrac{\pi}{6}$, $\pi$

010291
(1) 最大值为$3$, 取得最大值时$x$的集合为$\{x|x=k\pi, \ k\in \mathbf{Z}\}$; 最小值为$\dfrac{1}{3}$, 取得最小值时$x$的集合为$\{x|x=\dfrac{\pi}{2}+k\pi, \ k\in \mathbf{Z}\}$;\\
(2) 最大值为$1$, 取得最大值时$x$的集合为$\{x|x=2k\pi, \ k\in \mathbf{Z}\}$; 最小值为$-\dfrac{5}{4}$, 取最小值时$x$的集合为$\{x|x=\dfrac{2\pi}{3}+2k\pi\text{ 或 }\dfrac{4\pi}{3}+2k\pi, \ k\in \mathbf{Z}\}$

023607
在$[-\dfrac{\pi}{6},\dfrac{\pi}{3}]$上严格减, 在$[\dfrac{\pi}{3},\dfrac{2\pi}{3}]$上严格增, 值域为$[-1,1]$

018457
\begin{tikzpicture}[>=latex]
\draw [->] (-7,0) -- (7,0) node [below] {$x$};
\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
\draw (0,0) node [below right] {$O$};
\draw [domain = -7:7, samples = 200, ultra thick] plot (\x,{2*sin(\x/pi*180)});
\draw [domain = -7:7, samples = 200, thick] plot (\x,{1*sin(2*\x/pi*180)});
\draw [domain = -7:7, samples = 200, dashed, ultra thick] plot (\x,{sin(\x/pi*180+90)});
\draw [domain = -7:7, samples = 200, dashed] plot (\x,{sin(\x/pi*180)});
\draw [dashed] (-6,-3) -- (-5.5,-3) node [right] {$y=\sin x$};
\draw [ultra thick] (-3,-3) -- (-2.5,-3) node [right] {$y=2\sin x$};
\draw [thick] (0,-3) -- (0.5,-3) node [right] {$y=\sin 2x$};
\draw [ultra thick, dashed] (3,-3) -- (3.5,-3) node [right] {$y=\sin (x+\dfrac{\pi}{2})$};
\end{tikzpicture}

018458
大致图像: \begin{tikzpicture}[>=latex, scale = 0.5]
\draw [->] (-5,0) -- (5,0) node [below] {$x$};
\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};
\draw (0,0) node [below right] {$O$};
\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180+45)});
\foreach \i in {-3,-2,-1,1,2,3}
{\draw (0.1,\i) -- (0,\i) node [left] {$\i$};};
\foreach \i in {{-pi/8},{pi/8},{3*pi/8},{5*pi/8},{7*pi/8}}
{\draw (\i,0.2) -- (\i,0);};
\draw (-pi/8,0) node [above left] {$-\frac{\pi}{8}$};
\draw (pi/8,0) node [above] {$\frac{\pi}{8}$};
\draw (3*pi/8,0) node [above right] {$\frac{3\pi}{8}$};
\draw (5*pi/8,0) node [below] {$\frac{5\pi}{8}$};
\draw (7*pi/8,0) node [below right] {$\frac{7\pi}{8}$};
\end{tikzpicture}\\
振幅为$3$, 频率为$\dfrac{1}{\pi}$, 初始相位为$\dfrac{\pi}{4}$

018459
周期$T=0.02\text{s}$, 频率$f=50\text{Hz}$, 电流的最大值为$10\text{A}$; $I_0=10$, $\omega = 100\pi$, $\varphi = 0$

009607
(1) \begin{tikzpicture}[>=latex]
\draw [->] (-5,0) -- (5,0) node [below] {$x$};
\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw [domain = -5:5, samples = 100] plot (\x,{sin(\x/pi*180+30)});
\draw [dashed] (pi/3,0) node [below] {$\frac{\pi}{3}$}-- (pi/3,1) -- (0,1) node [left] {$1$};
\draw [dashed] (-2*pi/3,0) node [below] {$-\frac{2\pi}{3}$}-- (-2*pi/3,-1) -- (0,-1) node [right] {$-1$};
\end{tikzpicture}\\
(2) \begin{tikzpicture}[>=latex, scale = 0.5]
\draw [->] (-5,0) -- (5,0) node [below] {$x$};
\draw [->] (0,-4) -- (0,4) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180-60)});
\draw [dashed] (5*pi/12,0) node [below] {$\frac{5\pi}{12}$} --++ (0,3) -- (0,3) node [left] {$3$};
\draw [dashed] (-pi/12,0) node [above] {$\frac{\pi}{12}$} --++ (0,-3) -- (0,-3) node [right] {$-3$};
\end{tikzpicture}

009608
D

009609
$y=\dfrac{3}{2}\sin (2x+\dfrac{\pi}{6})$

018460
$\dfrac{4\pi}{3}$

018461
$\varphi=\dfrac{\pi}{2}$, $\omega = \dfrac{3}{4}$

010298
振幅为$\sqrt{2}$, 频率为$30\pi$, 初始相位为$-\dfrac{\pi}{12}$

010301
\begin{tikzpicture}[>=latex]
\draw [->] (-0.5,0) -- (3.5,0) node [below] {$x$};
\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw [domain = 0:3.5, samples = 100] plot (\x,{2*sin(\x*180+45)});
\draw [dashed] (0.25,0) node [below] {$\frac{1}{4}$} --++ (0,2) -- (0,2) node [left] {$2$};
\draw [dashed] (1.25,0) node [above] {$\frac{5}{4}$} --++ (0,-2) -- (0,-2) node [left] {$-2$};
\end{tikzpicture}\\
(1) 开始振动的位置在平衡位置上方$\sqrt{2}\text{cm}$处; (2) 最高点和最低点与平衡位置的距离都是$2$; (3) 经过$2\text{s}$小球往复振动一次; (4) 每秒小球往复振动$0.5$次

010303
$y=2\cos(\dfrac{1}{2} x+\dfrac{5\pi}{3})$

018462
定义域为$\{x|x\in \mathbf{R}, \ x\ne 6k+1, \ k\in \mathbf{Z}\}$, 单调增区间为$(6k-5,6k+1)$, $k\in \mathbf{Z}$

018463
$\dfrac{\pi}{2}$

018464
$2\sqrt{2}$

009610
$\{\alpha|\alpha = \dfrac{\pi}{3}+k\pi, \ k\in \mathbf{Z}\}$

009611
(1) $\tan (-\dfrac 27\pi )>\tan (-\dfrac 25\pi )$, 理由略; (2) $\cot 231^\circ>\cot 237^\circ$, 理由略; (3) $\tan (k\pi -\dfrac\pi 3)<\tan (k\pi +\dfrac\pi 3)$, 理由略

018465
$n=3$, 在$(-\dfrac{\pi}{18}+\dfrac{k\pi}{3}, \dfrac{5\pi}{18}+\dfrac{k\pi}{3})$, $k\in \mathbf{Z}$上是严格增函数

010308
(1) 奇函数, 理由略; (2) 偶函数, 理由略; (3) 奇函数, 理由略; (4) 偶函数, 理由略

024747
$\pi$

024748
面积$S=2\sqrt{2}\sin\theta\sin(\dfrac{\pi}{4}-\theta)(=\sqrt{2}\sin(2\theta+\dfrac{\pi}{4})-1)$, $\theta\in (0,\dfrac{\pi}{4})$, 面积的最大值为$\sqrt{2}-1$