ans

021441
错误, 正确, 错误, 错误


021442
D


021443
C


021444
A


021445
C


021446
D


021447
$-390^\circ$


021448
$304^\circ$, $-56^\circ$


021449
$-144^\circ$


021450
二, 四


021451
(1) $\{\alpha|\alpha=60^\circ+k\cdot 360^\circ, \ k\in \mathbf{Z}\}$, $-300^\circ$, $60^\circ$, $420^\circ$; (2) $\{\alpha|\alpha = -21^\circ+k\cdot 360^\circ, \ k \in \mathbf{Z}\}$, $-21^\circ$, $339^\circ$, $699^\circ$


021452
\begin{tikzpicture}[>=latex]
\fill [pattern = north east lines] (30:2) arc (30:60:2) -- (0,0) -- cycle;
\draw (30:2) -- (0,0) -- (60:2);
\draw [->] (-2,0) -- (2,0) node [below] {$x$};
\draw [->] (0,-2) -- (0,2) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\end{tikzpicture}


021453
$-1290^{\circ}$;第二象限


021454
(1) $ \{\alpha|\alpha=45^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\
(2) $\{\alpha|\alpha=135^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\
(3) $\{\alpha|\alpha=45^{\circ}+k\cdot 90^{\circ}, \ k \in \mathbf{Z}\}$;\\
(4) $\{\alpha|180^{\circ}+k\cdot 360^{\circ}<\alpha<270^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$.


021455
(1) $ \{\beta|\beta=\alpha+180^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\
(2) $\{\beta|\beta=\alpha+90^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\
(3) $\{\beta|\beta=-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\
(4) $\{\beta|\beta=90^{\circ}-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$.


021456
C


021457
B


021458
$\dfrac{\pi}{12}$; $\dfrac{7\pi}{12}$; $\dfrac{5\pi}{4}$; $300^{\circ}$; $324^{\circ}$; $315^{\circ}$; $(\dfrac{270}{\pi})^{\circ}$


021459
(1)$\frac{50\pi+180}{9}$;(2)$\frac{250\pi}{9}$


021460
$\sqrt{3}$


021461
(1)$\frac{\pi}{3}$;(2)$\frac{2\pi}{3}$


021462
(1)$16\pi+\frac{2\pi}{3}$,二;\\
(2)$-18\pi+\frac{4\pi}{3}$,三;\\
(3)$-2\pi+\frac{7\pi}{5}$,三;\\
(4)$-2\pi+\frac{3\pi}{4}$,二.


021463
$\frac{1}{2}$


021464
(1) $\{\alpha|-\frac{\pi}{2}+2k\pi<\alpha<2k\pi,\ k \in \mathbf{Z}\}$;\\
(2) $\{\alpha|\alpha=\frac{k\pi}{2},\ k \in \mathbf{Z}\}$.


021465
(1) $\beta=\alpha+2k\pi,\ k \in \mathbf{Z}$;\\
(2) $\beta=-\alpha+2k\pi,\ k \in \mathbf{Z}$;\\
(3) $\beta=-\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$;\\
(4) $\beta=\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$.


021466
(1) $\{\alpha|-\frac{\pi}{4}+2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\
(2) $\{\alpha|\frac{\pi}{6}+k\pi \le \alpha \le \frac{5\pi}{6}+k\pi,\ k \in \mathbf{Z}\}$.


021467
(1) 第四象限；第四象限；\\
(2) 第二象限或者第四象限；第一象限或第二象限或者$y$轴正半轴.


021468
$A\cap B=\{\alpha | 2k \pi+\dfrac{5\pi}{6}<\alpha<2k \pi+\dfrac{7\pi}{6},\ k \in \mathbf{Z} \}$


021469
\begin{tabular}{|c|c|c|c|c|c|}
\hline &$P(-5,12)$&$P(0,-6)$&$P(6,0)$&$P(-9,-12)$&$P(1,-\sqrt{3})$\\
\hline$\sin \alpha$&$\dfrac{12}{13}$ &$-1$ & $0$&$-\dfrac{4}{5}$ &$-\dfrac{\sqrt{3}}2$ \\
\hline$\cos \alpha$&$-\dfrac{5}{13}$ &$0$ & $1$&$-\dfrac{3}{5}$ &$\dfrac 12$ \\
\hline$\tan \alpha$&$-\dfrac{12}{5}$ &不存在 & $0$&$\dfrac{4}{3}$ &$-\sqrt{3}$ \\
\hline$\cot \alpha$&$-\dfrac{5}{12}$ &$0$ & 不存在 &$\dfrac {3}{4}$ &$-\dfrac{\sqrt{3}}3$ \\
\hline
\end{tabular}


021470
$2\sqrt{5}$


021471
$\frac{2\sqrt{13}}{13}$;$-\frac{2}{3}$


021472
$ \left( -2,\frac{2}{3} \right)$


021473
$<$


021474
5


021475
2


021476
当$t=\sqrt{5}$时, $\cos \alpha=- \frac{\sqrt{6}}{4}$, $\tan \alpha =- \frac{\sqrt{15}}{3}$;\\
当$t=-\sqrt{5}$时, $\cos \alpha=- \frac{\sqrt{6}}{4}$, $\tan \alpha = \frac{\sqrt{15}}{3}$;\\
当$t=0$时, $\cos \alpha=-1$, $\tan \alpha = 0$.


021477
当$\alpha$在第二象限时，$ \sin \alpha =\frac{4}{5}$, $\tan \alpha=-\frac{4}{3}$;\\
当$\alpha$在第三象限时，$ \sin \alpha =-\frac{4}{5}$, $\tan \alpha=\frac{4}{3}$.


021478
$-\frac{\sqrt{3}}{4}$


021479
(1) 第四象限； (2) 第一、四象限；(3)第一、三象限；(4)第一、三象限.


021480
$A=\left\{ -2,-0,4 \right\}$


021481
(1) $\{\alpha|2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\
(2) $[0,3)$


021482
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline$\alpha$&$\dfrac{\pi}{3}$&$\dfrac{7 \pi}{4}$&$\dfrac{2021 \pi}{2}$&$-\dfrac{\pi}{6}$&$-\dfrac{22 \pi}{3}$\\
\hline$\sin \alpha$&  $\frac{\sqrt{3}}{2}$ &$-\frac{\sqrt{2}}{2}$ & $1$&$-\frac{1}{2}$ &$\frac{\sqrt{3}}{2}$ \\
\hline$\cos \alpha$&$\frac{1}{2}$ &$\frac{\sqrt{2}}{2}$ & $0$&$\frac{\sqrt{3}}{2}$ &$-\frac{1}{2}$ \\
\hline$\tan \alpha$&$\sqrt{3}$ &$-1$ & 不存在 &$-\frac{\sqrt{3}}{3}$ &$-\sqrt{3}$\\
\hline$\cot \alpha$&$\frac{\sqrt{3}}{3}$ &$-1$  & $ 0$&$-\sqrt{3}$ &$-\frac{\sqrt{3}}{3}$ \\
\hline
\end{tabular}
\end{center}


021483
(1) $\{x|x=\frac{4\pi}{3}+2k \pi$或$ x=\frac{5\pi}{3}+2k \pi,\ k \in \mathbf{Z} \}$;\\
(2) $\{-\frac{2\pi}{3},-\frac{\pi}{3},\frac{4\pi}{3} ,\frac{5\pi}{3},\frac{10\pi}{3},\frac{11\pi}{3} \}$


021484
$-\frac{2\sqrt{5}}{5}$;$2$


021485
\textcircled{2} \textcircled{4}


021486
当$\alpha$在第一象限时，$ \sin \alpha =\frac{3\sqrt{10}}{10}$, $\cos \alpha =\frac{\sqrt{10}}{10}$,$\tan \alpha=3$;\\
当$\alpha$在第三象限时，$ \sin \alpha =-\frac{3\sqrt{10}}{10}$,$\cos \alpha =-\frac{\sqrt{10}}{10}$, $\tan \alpha=3$.


021487
$\sin k\pi =0$; $\cos k\pi=\begin{cases}1, & k=2n, \\ -1, & k=2n-1\end{cases}$($n \in \mathbf{Z}$).


021488
(1) $\{\theta | 2k \pi+\dfrac{\pi}{3}<\theta<2k \pi+\dfrac{2\pi}{3},\ k \in \mathbf{Z} \}$;\\
(2) $\{\theta | k \pi-\dfrac{\pi}{2}<\theta \le k \pi-\dfrac{\pi}{6},\ k \in \mathbf{Z} \}$;\\
(3) $\{\theta | k \pi+\dfrac{\pi}{3} \le \theta \le k \pi+\dfrac{2\pi}{3},\ k \in \mathbf{Z} \}$.


021489
第二象限


021490
(1) 当$\dfrac{\alpha}{2}$在第二象限时，点$P$在第四象限；\\
当$\dfrac{\alpha}{2}$在第四象限时，点$P$在第二象限.\\
(2) $\sin (\cos \alpha) \cdot \cos (\sin \alpha)<0$


021491
当$m=0$时，$ \cos (\alpha+1905^{\circ})=-1$,$\tan (\alpha-615^{\circ})=0$;\\
当$m=\sqrt{5}$时，$ \cos (\alpha+1905^{\circ})  =-\frac{\sqrt{6}}{4}$,$\tan (\alpha-615^{\circ})=-\frac{\sqrt{15}}{3}$;\\
当$m=-\sqrt{5}$时，$ \cos (\alpha+1905^{\circ})  =-\frac{\sqrt{6}}{4}$,$\tan (\alpha-615^{\circ})=\frac{\sqrt{15}}{3}$.


021492
$-\dfrac{3}{8}$


021493
$-\dfrac{1}{20}$


021494
$\dfrac{7\sqrt{2}}{4}$


021495
$\dfrac{3\sqrt{5}}{5}$


021496
$11$


021497
$5$;$-\dfrac{12}{5}$;$\dfrac{4}{9}$


021498
$\sin ^2 \alpha$


021499
$1$


021502
$-\dfrac{12}{5}$


021503
$-\dfrac{\sqrt{3}}{2}$


021504
$\dfrac{\sqrt{7}}{2}$;$\dfrac{\sqrt{7}}{4}$


021505
$-\dfrac{\sqrt{11}}{3}$


021506
$\dfrac{\pi}{3}$


021507
$\left[ 0,\pi \right )$


021508
$-\dfrac{\sqrt{3}}{2}$;$-\dfrac{\sqrt{2}}{2}$;$-\sqrt{3}$;$-\sqrt{3}$


021509
$69^{\circ}$;$72^{\circ}$;$\dfrac{\pi}{9}$;$\dfrac{7 \pi}{15}$


021510
$\cot \alpha$


021511
$-1$


021512
$-1$


021513
$ \sin 2-\cos 2$


021514
$0$


021515
$0$


021516
$-\dfrac{\sqrt{1-a^2}}{a}$


021517
$-\dfrac{2+\sqrt{3}}{3}$


021518
(1) $\dfrac{\sqrt{3}}{2}$;(2) $\dfrac{1}{4}$.


021519
(1) $-\dfrac{2}{3}$; \\
(2) $\dfrac{2}{3}$; \\
(3) $-\dfrac{\sqrt{5}}{3}$;\\
(4) $\dfrac{\sqrt{5}}{2}$.


021520
(1) $\sin 69^{\circ}$ ; (2) $-\cos 8^{\circ}$ ;  
(3) $-\tan  \dfrac{\pi}{9}$; (4) $\cot  \dfrac{7\pi}{15}$.


021521
$\dfrac{2}{5}$


021522
$(3,4)$


021523
$0$


021524
$\sin \alpha$


021525
$-\dfrac{1}{5}$


021526
(1) $\dfrac{\sqrt{6}}{6}-\sqrt{3}$;\\
(2) $-\dfrac{\sqrt{6}}{3}$;\\
(3) $1$


021527
(1) $\dfrac{6 \pi}{5}$; (2) $\dfrac{4 \pi}{5}$; (3) $\dfrac{13 \pi}{10}$; (4) $\dfrac{17 \pi}{10}$.


021528
(1) 当$\alpha$在第一象限时, $\sin (2 \pi-\alpha)=-\dfrac{\sqrt{3}}{2}$;
当$\alpha$在第三象限时, $\sin (2 \pi-\alpha)=\dfrac{\sqrt{3}}{2}$.\\
(2) 当$\alpha$在第一象限时, $\dfrac{1}{\tan [\dfrac{(2 k+1) \pi}{2}+\alpha]}=-\sqrt{3}$;
当$\alpha$在第四象限时, $\dfrac{1}{\tan [\dfrac{(2 k+1) \pi}{2}+\alpha]}=\sqrt{3}$.


021529
(1)  $\{x | x=k \pi+ (-1)^k \cdot \dfrac{\pi}{4},\ k \in \mathbf{Z}\}$;\\
(2) $\{x | x=2k \pi \pm  \dfrac{2\pi}{3},\ k \in \mathbf{Z}\}$;\\
(3) $\{x | x=k \pi +  \dfrac{5\pi}{6},\ k \in \mathbf{Z}\}$;\\
(4) $\{x | x=2k \pi +  \dfrac{5\pi}{6}$ 或$x=2k \pi + \dfrac{3\pi}{2} ,\ k \in \mathbf{Z}\}$;\\
第二种写法： $\{x | x=k \pi+ (-1)^k \cdot \dfrac{\pi}{6}+\dfrac{2\pi}{3},\ k \in \mathbf{Z}\}$;\\
(5)  $\{x | x=k \pi - \arctan \dfrac{\sqrt{3}}{2}+  \dfrac{\pi}{4},\ k \in \mathbf{Z}\}$;\\
(6)  $\{x | x=\dfrac{2k \pi}{5} +  \dfrac{7\pi}{60}$ 或$ x=\dfrac{2k \pi}{5} - \dfrac{13\pi}{60} ,\ k \in \mathbf{Z}\}$;\\
(7)  $\{x | x=k \pi - \dfrac{5\pi}{8}$ 或$x=k \pi -  \dfrac{3\pi}{8} ,\ k \in \mathbf{Z}\}$;


021530
(1) $\{ \dfrac{\pi}{12},\dfrac{17\pi}{12}  \}$;\\
(2) $\{ \dfrac{5\pi}{6} \}$;\\
(3) $\{ \dfrac{\pi}{12},\dfrac{5\pi}{12}  \}$;\\
(4) $\{ \dfrac{5\pi}{6} \}$.


021531
(1)  $\{x | x= \dfrac{2k \pi}{5} ,\ k \in \mathbf{Z}\}$;\\
(2)  $\{x | x= \dfrac{2k \pi}{3} +\dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$;\\
(3)  $\{x | x= 2k \pi$ 或$x=k \pi  +(-1)^k \cdot \dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$;\\
(4)  $\{x | x= k \pi+\dfrac{ \pi}{3}$ 或$x=k \pi -\dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$.


021532
$\dfrac{3+4\sqrt{3}}{10}$


021533
$-1$


021534
$-\dfrac{33}{50}$


021535
(1) $\dfrac{\sqrt{6}-\sqrt{2}}{4}$;
(2) $\dfrac{\sqrt{6}+\sqrt{2}}{4}$;
(3) $0$.


021536
(1) $\sqrt{3} \sin \alpha$;
(2) $\cos(\alpha-2\beta)$.


021537
$\dfrac{140}{221}$


021538
$\dfrac{2\sqrt{6}-1}{6}$


021539
证明略


021540
C


021541
A


021542
$\dfrac{3\sqrt{10}+6\sqrt{2}+2\sqrt{14}-\sqrt{70}}{24}$


021543
$\dfrac{8\sqrt{3}-21}{20}$


021544
$\dfrac{\pi}{2}$


021545
$-\dfrac{2+\sqrt{15}}{6}$


021546
$-\dfrac{\sqrt{2}}{2}$


021547
$\sin 2\beta$


021548
$0$


021549
$2-\sqrt{3}$


021550
$\dfrac{16}{65}$


021551
$-\dfrac{7}{25}$


021552
$-\dfrac{4\sqrt{14}+3\sqrt{2}}{20}$


021553
$(\dfrac{4\sqrt{3}+3}{2},\dfrac{3\sqrt{3}-4}{2})$


021554
$-\dfrac{33}{65}$或$\dfrac{63}{65}$


021555
B


021556
C


021557
$-\dfrac{56}{65}$


021558
$-3$


021559
$\dfrac{3}{4}$


021560
$\dfrac{-6+5\sqrt{3}}{3}$


021561
$\tan \alpha$


021562
$\sqrt{3}$


021563
$-\dfrac{\sqrt{3}}{3}$


021564
A


021565
$-\dfrac{17}{31}$


021566
$\dfrac{\pi}{4}$


021567
(1) $\dfrac{1}{3}$;
(2) $\dfrac{1}{7}$


021568
$-\dfrac{1}{5}$


021569
当$CD = 1.4$米时，$\tan \angle ACB$最大


021570
(1) $2 \sin (\alpha+\dfrac{\pi}{6})$;
(2) $\sqrt{2} \sin (\alpha+\dfrac{7\pi}{4})$.


021571
$6\cos(\alpha+\dfrac{\pi}{3})$


021572
$2k \pi-\dfrac{\pi}{3}(k\in \mathbf{Z}  )$


021573
B


021574
$\dfrac{1}{3}$


021575
$\dfrac{\pi}{12}$或$\dfrac{5\pi}{12}$


021576
$5$


021577
$\dfrac{13}{3}$


021578
$-\dfrac{p}{1+q}$


021579
$\dfrac{3}{5}$


021580
$\dfrac{24}{7}$


021581
$-\dfrac{24}{25}$


021582
$-\dfrac{15}{17}$


021583
$\sin 2 \varphi=\dfrac{4\sqrt{2}}{9}$;
$\cos 2 \varphi=-\dfrac{7}{9}$;
$\tan 2 \varphi=-\dfrac{4\sqrt{2}}{7}$.


021584
$\dfrac{24}{25}$; $\dfrac{7}{25}$; $\dfrac{24}{7}$


021585
(1) $-\dfrac{\sqrt{3}}{3}$;\\
(2) $\dfrac{3}{4}$.


021586
$\dfrac{7}{24}$


021587
$-\dfrac{2\sqrt{10}}{5}$


021588
$1$


021590
$1$或$\dfrac{7}{25}$


021591
$-\dfrac{\sqrt{2-2a}}{2}$


021592
第三象限


021593
当$\dfrac{\theta}{2}$在第二象限时，
$\sin \dfrac{\theta}{2}=\dfrac{\sqrt{3}}{3}$,
$\cos \dfrac{\theta}{2}=-\dfrac{\sqrt{6}}{3}$,
$\tan \dfrac{\theta}{2}=-\dfrac{\sqrt{2}}{2}$;\\
当$\dfrac{\theta}{2}$在第四象限时，
$\sin \dfrac{\theta}{2}=-\dfrac{\sqrt{3}}{3}$,
$\cos \dfrac{\theta}{2}=\dfrac{\sqrt{6}}{3}$,
$\tan \dfrac{\theta}{2}=-\dfrac{\sqrt{2}}{2}$.


021594
$\dfrac{3}{5}$;$\dfrac{4}{5}$


021596
$\dfrac{2}{3}$


021597
$\cos \alpha-\sin \alpha$


021598
$\sin \dfrac{ \alpha}{2}$


021599
(1) $\tan \dfrac{\theta}{2}$; (2) $\sin \alpha$.


021600
$\dfrac{\sqrt{6}}{2}$


021601
$30^{\circ}$或$90^{\circ}$


021602
$\sqrt{6}$


021603
$55$


021604
$\dfrac{\pi}{3}$或$\dfrac{2\pi}{3}$


021605
$1: \sqrt{3}: 2$


021606
$2$


021607
$\dfrac{5}{8}$


021608
等腰


021609
$\dfrac{3\sqrt{2}}{2}$


021610
$\sqrt{3}$


021611
$\dfrac{7\pi}{12}$


021612
$\dfrac{2\pi}{3}$


021613
(1) $\left( 0,9 \right)$; \\
(2) $\{9\} \cup \left[18,+ \infty \right)$;\\
(3) $\left( 9,18 \right)$.


021614
$\dfrac{3\sqrt{7}}{8}$


021615
$\sqrt{17}$或$\sqrt{65}$


021616
$\dfrac{\pi}{4}$


021617
\textcircled{1}；\textcircled{2}


021618
$a>3$


021619
$a=\sqrt{21}$和$\sin B=\dfrac{5\sqrt{7}}{14}$


021620
$\dfrac{2\pi}{3}$


021621
$c=\sqrt{6}+\sqrt{2}$;$C=75^\circ$.


021622
$\dfrac{\sqrt{19}}{2}$


021623
周长的最小值为$12$,此时三角形为正三角形；\\
面积最大值为$4\sqrt{3}$,此时三角形为正三角形.


021624
$\dfrac{\sqrt{5}}{5}$


021625
$\dfrac{2\sqrt{5}}{5}$或$-\dfrac{2\sqrt{5}}{25}$


021626
$\dfrac{\sqrt{5}}{5}$或$\dfrac{11\sqrt{5}}{25}$


021627
$\left ( 2,2\sqrt{2} \right )$


021628
(1) 以$C$为直角的直角三角形；\\
(2) 以$A$为顶角的等腰三角形；\\
(3)  以$A$为直角的直角三角形.


021629
$a=\sqrt{13}$;$R=\dfrac{\sqrt{39}}{3}$.


021630
$6\sqrt{19}$


021631
(1) $x=\arcsin \dfrac{2}{5}$或$\pi-\arcsin \dfrac{2}{5}$;\\
(2) $x=\pi-\arccos \dfrac{2}{3}$或$\pi+\arccos \dfrac{2}{3}$;\\
(3) $x=k\pi- \arctan \dfrac{1}{2},k \in \mathbf{Z}$.


021632
$300\sqrt{3}$


021633
证明略


021634
$\theta=\dfrac{\pi}{12}$;塔高为$1.5$千米.


021635
$64.81$米


021636
(1) $3.9$千米;(2) $4.0$千米.


021637
$2.4$千米


021638
$\dfrac{\pi}{2}$


021639
B


021640
(1) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (-4,0) -- (4,0) node [below] {$x$};
\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};
\draw (0,0) node [below right] {$O$};
\draw (-pi,0.1) -- (-pi,0) node [below left] {$-\pi$};
\draw (-0.5*pi,0.1) -- (-0.5*pi,0) node [below] {$-\frac{\pi}{2}$};
\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\frac{\pi}{2}$};
\draw (pi,0.1) -- (pi,0) node [below] {$\pi$};
\draw (0.1,1) -- (0,1) node [left] {$1$};
\draw (0.1,-1) -- (0,-1) node [left] {$-1$};
\draw [domain = -pi:pi,samples = 100] plot (\x,{sin(\x/pi*180)+1});
\end{tikzpicture}\\
(2) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (0,0) -- (7,0) node [below] {$x$};
\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};
\draw (0,0) node [below right] {$O$};
\draw (pi/2,0.1) -- (pi/2,0) node [below] {$\frac{\pi}{2}$};
\draw (pi,0.1) -- (pi,0) node [below] {$\pi$};
\draw (1.5*pi,0.1) -- (1.5*pi,0) node [below] {$\frac{3\pi}{2}$};
\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\pi$};
\draw (0.1,1) -- (0,1) node [left] {$1$};
\draw (0.1,-1) -- (0,-1) node [left] {$-1$};
\draw [domain = 0:2*pi,samples = 100] plot (\x,{-cos(\x/pi*180)});
\end{tikzpicture}


021641
(1) 定义域为$\left \{x|x \neq-\dfrac{\pi}{2}+2k\pi,k \in \mathbf{Z} \right \}$;\\
(2) 定义域为$\left \{x|\dfrac{\pi}{2}+2k\pi \leq x \leq \dfrac{3\pi}{2}+2k\pi,k \in \mathbf{Z} \right \}$.


021642
$\left \{x|\dfrac{\pi}{6} \leq x \leq \dfrac{5\pi}{6},k \in \mathbf{Z} \right \}$


021643
$2\pi$


021644
C


021645
C


021646
(1) 当$a \in (-\infty，-\dfrac{\sqrt{2}}{2})\cup (1,+\infty)$ 时，方程实数解个数为$0$个；\\
当$a \in [-\dfrac{\sqrt{2}}{2},0)\cup \{1\}$ 时，方程实数解个数为$1$个；\\
当$a \in [0,1)$时，方程实数解个数为$2$个.\\
(2) 当$a \in (-\infty，-1)\cup (1,+\infty)$ 时，方程实数解个数为$0$个；\\
当$a \in (0,1]$时，方程实数解个数为$1$个;\\
当$a \in \{0,-1\}$时，方程实数解个数为$2$个;\\
当$a \in (-1,0)$时，方程实数解个数为$3$个.


021647
(1) $8\pi$;
(2) $\pi$;(3) $\pi$;(4) $2\pi$.


021648
$3$


021649
A


021650
C


021651
(1) 假;(1) 假;(3) 真.


021652
D


021653
(1) $\pi$; (2) $\pi$; (3) $\dfrac{\pi}{2}$; (4) $\dfrac{\pi}{|a|}$.


021654
$4\sin(\dfrac{\pi x}{2})-2$


021655
B


021656
A


021657
(1) $f(3)=-1$; $f(5)=1$; $f(7)=-1$;\\
(2) $T=4$.


021658
$\left [2,4\right] $


021659
$\left [-2,2\right] $


021660
$ [-\dfrac{3}{2},3] $


021661
$ (-\dfrac{\sqrt{3}}{2},1] $


021662
$3$; $\left \{x|x=-\dfrac{\pi}{2}+2k\pi,k \in \bf{Z} \right\}$


021663
$-3$; $\left \{x|x=-\dfrac{\pi}{12}+k\pi,k \in \bf{Z} \right\}$


021664
当$x=\dfrac{\pi}{2}-\arcsin \dfrac{3\sqrt{13}}{13}+2k\pi,k \in \bf{Z}$时，函数的最大值为$\sqrt{13}$;\\
当$x=-\dfrac{\pi}{2}-\arcsin \dfrac{3\sqrt{13}}{13}+2k\pi,k \in \bf{Z}$时，函数的最小值为$-\sqrt{13}$.


021665
D


021666
C


021667
当$\alpha=\dfrac{\pi}{2}-\theta$时，竹竿的影子最长，最长为$\dfrac{\sin(\alpha+\theta)}{\sin \theta}*l$.


021668
$[-1,1]$


021669
$\{x|x\neq 2k\pi,k \in \bf{Z}\}$;$(-\infty,0]$


021670
$k=3$或$-3$;$b=-1$


021671
当$x=0$时，函数$y$取到最大值，最大值为$0$;\\
当$x=\dfrac{\pi}{4}$时，函数$y$取到最小值，最小值为$-1$.


021672
$f(a)=\begin{cases}
a^2+2a+2, & a\leq -1,\\
1, & -1<a<1,\\
a^2-2a+2, & a\geq 1.
\end{cases}$


021673
(1) 偶函数； (2) 不是奇函数也不是偶函数； (3) 奇函数； (4) 偶函数.


021674
(1) 真命题； (2) 真命题； (3) 假命题； (4) 真命题.


021675
B


021676
$f(x)=\cos4x$


021677
$-7$


021678
(1) 不是奇函数也不是偶函数；(2) 偶函数；(3) 偶函数.


021679
(1) 假命题；(2) 假命题；(3) 真命题； (4) 真命题.


021680
不存在这样的$\theta$,使得函数$f(x)=1+\sin (x+\theta)+\sqrt{3} \cos (x+\theta)$为奇函数.


021681
$0<\beta<\alpha<\dfrac{\pi}{2}$


021682
(1) 假命题； (2) 真命题；(3) 假命题； (4)真命题.


021683
$[-\dfrac{3\pi}{2},-\dfrac{\pi}{2}]$


021684
$[-\dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+k\pi],k \in \bf{Z}$


021685
(1) $[\dfrac{3\pi}{2},2\pi]$;
(2) $[\pi,\dfrac{3\pi}{2}]$.


021686
(1) $[\dfrac{5\pi}{4}+2k\pi,\dfrac{9\pi}{4}+2k\pi],k \in \bf{Z}$;\\
(2) $[-\dfrac{7\pi}{6}+2k\pi,-\dfrac{\pi}{6}+2k\pi],k \in \bf{Z}$.


021687
$[\dfrac{\pi}{6}+\dfrac{k\pi}{2},\dfrac{5\pi}{12}+\dfrac{k\pi}{2}],k \in \bf{Z}$


021688
$[k\pi,k\pi+\dfrac{\pi}{2}],k \in \bf{Z}$


021690
$4\pi$; $3$; $\dfrac{\pi}{3}$; $\dfrac{1}{4\pi}$.


021691
$\pi$; $[-\dfrac{4}{3},\dfrac{4}{3}].$


021692
$b-|a|$


021693
$y=3\sin(7x+\dfrac{\pi}{6})$


021694
(1) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (-4,0) -- (4,0) node [below] {$x$};
\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw ({-pi/12},0.1) -- ({-pi/12},0) node [below left] {$-\frac{\pi}{12}$};
\draw ({pi/6},0.1) -- ({pi/6},0) node [below] {$\frac{\pi}{6}$};
\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\frac{5\pi}{12}$};
\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above] {$\frac{2\pi}{3}$};
\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\frac{11\pi}{12}$};
\draw (0.1,1) -- (0,1) node [left] {$1$};
\draw (0.1,-1) -- (0,-1) node [left] {$-1$};
\draw [domain = {-pi/12}:{11*pi/12},samples = 100] plot (\x,{sin(2*\x/pi*180+30)});
\end{tikzpicture}\\
(2) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (-1,0) -- (15,0) node [below] {$x$};
\draw [->] (0,-3) -- (0,3) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\pi$};
\draw (pi,0.1) -- (pi,0) node [below] {$\pi$};
\draw (3*pi,0.1) -- (3*pi,0) node [below] {$\frac{3\pi}{2}$};
\draw (4*pi,0.1) -- (4*pi,0) node [below] {$4\pi$};
\draw (0.1,2) -- (0,2) node [left] {$2$};
\draw (0.1,-2) -- (0,-2) node [left] {$-2$};
\draw [domain =0:4*pi,samples = 100] plot (\x,{2*sin(0.5*\x/pi*180)});
\end{tikzpicture}\\
(3) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (-1,0) -- (4,0) node [below] {$x$};
\draw [->] (0,-1) -- (0,1) node [left] {$y$};
\draw (0,0) node [below right] {$O$};
\draw (0.25*pi,0.1) -- (0.25*pi,0) node [below] {$\frac{\pi}{4}$};
\draw (pi,0.1) -- (pi,0) node [below] {$\pi$};
\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\frac{\pi}{2}$};
\draw (0.75*pi,0.1) -- (0.75*pi,0) node [above] {$\frac{3\pi}{4}$};
\draw (0.1,0.5) -- (0,0.5) node [left] {$\frac{1}{2}$};
\draw (0.1,-0.5) -- (0,-0.5) node [left] {$-\frac{1}{2}$};
\draw [domain =0:pi,samples = 100] plot (\x,{0.5*sin(2*\x/pi*180)});
\end{tikzpicture}\\
(4) \begin{tikzpicture}[>=latex, scale = 0.7]
\draw [->] (-1.5,0) -- (3.5,0) node [below] {$x$};
\draw [->] (0,-5.5) -- (0,5.5) node [left] {$y$};
\draw (0,0) node [below left] {$O$};
\draw ({-pi/3},0.1) -- ({-pi/3},0) node [below left] {$-\frac{\pi}{3}$};
\draw ({-pi/12},0.1) -- ({-pi/12},0) node [above left] {$-\frac{\pi}{12}$};
\draw ({pi/6},0.1) -- ({pi/6},0) node [below right] {$\frac{\pi}{6}$};
\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\frac{5\pi}{12}$};
\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above right] {$\frac{2\pi}{3}$};
\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\frac{11\pi}{12}$};
\draw (0.1,5) -- (0,5) node [left] {$5$};
\draw (0.1,-5) -- (0,-5) node [below left] {$-5$};
\draw [domain = {-4*pi/12}:{2*pi/3},samples = 100] plot (\x,{5*sin(2*\x/pi*180-60)});
\end{tikzpicture}


021695
$4\pi$;$4$.


021696
$f(x)=4\sin(x+\dfrac{\pi}{6})$


021697
(1) $f(x)=\dfrac{\sqrt{3}}{2}\sin(3x+\pi)+\dfrac{\sqrt{3}}{2};$\\
(2) $[-\dfrac{\pi}{2}+\dfrac{2k\pi}{3},-\dfrac{\pi}{6}+\dfrac{2k\pi}{3}],k \in \bf{Z}$;\\
(3) 函数最大值为$\sqrt{3}$,此时$x$值为${x|x=-\dfrac{\pi}{6}+\dfrac{2k\pi}{3},k \in \bf{Z}}$


021698
$x=\pi+2k\pi,k \in \bf{Z}$


021699
纵;伸长; $3$.


021700
缩短; $\dfrac{1}{2}$; 缩短; $\dfrac{1}{3}$.


021701
$f(x)=\sin(\dfrac{1}{2}x-\dfrac{\pi}{3})$


021702
$f(x)=\sin(\dfrac{1}{2}x-\dfrac{\pi}{6})$


021703
$f(x)=2\sin(\dfrac{1}{3}x+\dfrac{\pi}{6})$


021704
$x=\dfrac{\pi}{3}+2k\pi,k \in \bf{Z}$; $(-\dfrac{2\pi}{3}+2k\pi,0),k \in \bf{Z}$.


021705
C


021706
左; $\dfrac{\pi}{8}$.


021707
$f(x)=\sin(2x+\dfrac{\pi}{2})$,
$g(x)=\sin x$.


021708
(1) $\sqrt{2}$;
(2) $g(x)=2\cos(\dfrac{1}{2}x-\dfrac{\pi}{3}) $, 单调递减区间为$[\dfrac{2\pi}{3}+4k\pi,\dfrac{8\pi}{3}+4k\pi],k \in \bf{Z}$.


021709
(1) $2\pi$; (2) $1$; (3) $\dfrac{\pi}{2}$.


021710
(1) $[0,\dfrac{\pi}{2})$, $(\dfrac{3\pi}{2},2\pi]$; \\
(2) $[0,\dfrac{\pi}{2})$, $(\dfrac{\pi}{2},\pi]$.


021711
(1) 奇函数; (2) 偶函数.


021712
$[-5,+\infty)$


021713
(1) $<$; (2) $>$; (3) $>$;  (4)$<$.


021714
\textcircled{3}


021715
最小值为$-\dfrac{\sqrt{3}}{3}$，此时$x=-\dfrac{\pi}{3}$.


021716
(1) $ \{x|x \neq \dfrac{k\pi}{2},k \in \bf{Z}\} $;\\
(2) 单调增区间为$(-\dfrac{\pi}{2}+\dfrac{k\pi}{2},\dfrac{k\pi}{2}), k \in \bf{Z}$.


021717
$\{x|x\neq \dfrac{\pi}{4}-\dfrac{1}{2}+\dfrac{k\pi}{2},k \in \bf{Z} \}$


021718
$(-\dfrac{\pi}{4}+\dfrac{k\pi}{3},\dfrac{\pi}{12}+\dfrac{k\pi}{3}), k \in \bf{Z}$


021719
B


021720
定义域为$ \{x|x \neq \dfrac{7\pi}{5}+2k\pi,k \in \bf{Z}\} $;\\
严格增区间为$(-\dfrac{3\pi}{5}+2k\pi,\dfrac{7\pi}{5}+2k\pi), k \in \bf{Z}$.


021721
函数零点为$x=\dfrac{2k\pi}{5}+2k\pi,k \in \bf{Z}$.


021722
(1) 假命题; (2) 假命题; (3) 假命题; (4) 真命题.


021723
$[-4,2+4\sqrt{3}]$


021724
最大张角的正切值为$\dfrac{\sqrt{2}}{4}$, 此时学生距离时钟$\sqrt{0.18}$米.


021726
A


021727
C


021728
B


021729
单位圆


021730
B


021731
$\overrightarrow{CD}$


021732
$\overrightarrow{AC}$


021733
(1) 假命题; (2) 真命题; (3) 假命题; (4) 假命题.


021734
(1) $\overrightarrow{DB}$; $\overrightarrow{FE}$.\\
(2) $\overrightarrow{ED}$; $\overrightarrow{CF}$;  $\overrightarrow{FA}$.\\
(3) $\overrightarrow{EF}$; $\overrightarrow{AD}$; $\overrightarrow{DA}$; $\overrightarrow{DB}$; $\overrightarrow{BD}$; $\overrightarrow{AB}$; $\overrightarrow{BA}$.


021735
$40$


021736
$40$


021737
$2$


021739
$-3\overrightarrow {a}+6 \overrightarrow {b}$


021740
$7 \overrightarrow {a}-2 \overrightarrow {b}- \overrightarrow {c}$


021741
(1) 假命题; (2) 真命题; (3) 假命题; (4) 真命题.


021742
(1) $\overrightarrow {AB}=\dfrac{1}{2}\overrightarrow {a}-\dfrac{1}{2}\overrightarrow {b}$;\\
(2) $\overrightarrow {BC}=\dfrac{1}{2}\overrightarrow {a}+\dfrac{1}{2}\overrightarrow {b}$.


021743
$\lambda=\dfrac{1}{3}$


021744
$x=2$; $y=1$.


021745
(2) $m=1$或$-1$.


021746
$\overrightarrow{DC}=\dfrac{1}{2}\overrightarrow{a}$;\\ $\overrightarrow{DC}=-\dfrac{1}{2}\overrightarrow{a}+\overrightarrow{b}$;\\
$\overrightarrow{MN}=-\dfrac{1}{4}\overrightarrow{a}-\overrightarrow{b}$.


021747
$\overrightarrow{0}$


021748
$\dfrac{2}{3}\overrightarrow{a}+\dfrac{1}{3}\overrightarrow{b}$


021749
A


021750
B


021751
C


021752
$\sqrt{3}$


021753
$-\dfrac{3\sqrt{3}}{2}$


021754
等边三角形


021755
$\dfrac{\pi}{4}$


021756
$\dfrac{2\pi}{3}$


021757
$-10\sqrt{2}$


021758
$\dfrac{4}{3}$


021759
$-\dfrac{2}{3}\overrightarrow {a}$


021760
B


021761
B


021762
A


021763
$7$


021764
$2$


021765
C


021766
外心; 重心; 垂心.


021767
$\dfrac{\pi}{3}$


021768
$-25$


021769
$\lambda=\dfrac{7}{12}$


021770
$AB=8$


021771
$t=\dfrac{1}{3}$


021772
(1) $(\overrightarrow {a}-\overrightarrow {b}) \cdot \overrightarrow {c}=\overrightarrow {a} \cdot \overrightarrow {c}- \overrightarrow {b} \cdot \overrightarrow {c}=1*1*(-\dfrac{1}{2})-1*1*(-\dfrac{1}{2})=0;\\$
(2) $k<0$或$k>2$.


021773
$[2,5]$


021774
$\arccos \dfrac{4}{5}$


021775
$\overrightarrow{OP}=\dfrac{3}{11}\overrightarrow {a}+\dfrac{2}{11}\overrightarrow {b}$


021776
(1) $(-1,0)$; (2) $(2,\dfrac{1}{2})$; (3) $(2,0)$或 $(-2,0)$; (4) $(\dfrac{3\sqrt{2}}{2},-\dfrac{3\sqrt{2}}{2})$.


021777
(1) 10; (2) $(-\dfrac{4}{5},\dfrac{3}{5})$.


021778
$x=4$, $y=1$.


021779
$(\dfrac{3}{5},-\dfrac{4}{5})$


021780
$(4,-8)$


021781
$(1,2)$


021782
C


021783
A


021784
B


021786
$\lambda=\mu$ 且$\lambda$和$\mu$非零.


021787
(1) 当$t=\dfrac{3}{2}$时,点$P$在$x$轴上; 当$t=\dfrac{1}{3}$时,点$P$在$y$轴上;当$-\dfrac{2}{3}<t<-\dfrac{1}{3}$时,点$P$在第二象限;\\
(2) 四边形$OABP$不能构成平行四边形.


021788
(1) $x+2y=0$; (2) 当$x=2,y=-1$时，四边形面积为$16$;当$x=-6,y=3$时，四边形面积为$16$.


021789
$(-1,-\dfrac{3}{2})$


021791
(1) $P(-\dfrac{5}{3},-\dfrac{5}{3})$; (2) $(-18,-4)$.


021792
$\overrightarrow {a} \parallel \overrightarrow {c}$,$\overrightarrow {b}  \parallel \overrightarrow {d}$.


021793
$(\dfrac{3}{5},\dfrac{4}{5})$


021794
B


021795
$\overrightarrow{PM}=-\dfrac{1}{3}\overrightarrow{b}+\dfrac{1}{3}\overrightarrow{c}$, $\overrightarrow{QB}=-\dfrac{2}{3}\overrightarrow{b}+\overrightarrow{c}$.


021796
$(2,2)$, $(-6,0)$,$(4,6)$.


021797
(1) $C(0,3)$、$D(3,0)$;\\
(2) $\overrightarrow{BD}=(4,-4)$.


021798
$(-2,5)$或$(6,-7)$


021799
$[-2,\dfrac{1}{4}]$


021800
(1) $-33$; (2) $2\sqrt{65}$.


021801
$\dfrac{\pi}{4}$


021802
不存在这样的点C.


021803
$(6,4)$或$(-6,-4)$.


021804
$(-2,1)$


021805
$(0,4)$或$(0,-2)$


021806
$2$或$4$


021807
A


021808
B


021809
$D$的坐标为$(1,1)$,$\overrightarrow{AD}=(-1,2)$.


021810
$\overrightarrow{OD}=(11,6)$


021811
以$C$为直角顶点的等腰直角三角形


021812
$(-\dfrac{10}{3},\dfrac{6}{5}) \cup (\dfrac{6}{5},+\infty)$


021813
$-\dfrac{4}{9}$


021814
$7$


021815
$\lambda=\dfrac{5}{17}$和$y=\dfrac{49}{22}$


021816
$(-2,-\dfrac{1}{5})$或$(10,-5)$


021817
$(\dfrac{5}{3},\dfrac{4}{3})$


021818
$(-1,-1)$


021819
$(\dfrac{7}{2},-\dfrac{3}{2})$或$(\dfrac{3}{2},\dfrac{7}{2})$


021820
$\overrightarrow {c}=\dfrac{2}{3}\overrightarrow {a}+\dfrac{1}{3}\overrightarrow {b}$


021821
$17$


021823
$x=1+\dfrac{\sqrt{3}}{2}$,$y=\dfrac{\sqrt{3}}{2}$.


021826
$\dfrac{13}{2}$


021827
$\overrightarrow{OG}=\dfrac{1}{3}(\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c})$


021828
(1) 当$x=\dfrac{43}{29}$时，最大值为$\sqrt{319}$;\\
(2) 当$x=2$时，最小值为$4*\sqrt{2}$;\\
(3) 当$x=\dfrac{2}{3}$时，最大值为$5$.


021829
B


021830
(1) $\dfrac{11}{12}-\dfrac{1}{10} \mathrm{i}$; (2) $-3$; (3) $2b+2a \mathrm{i}$.


021831
(1) $-\mathrm{i}$; (2) $145$; (3) $\mathrm{i}$; (4) $\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} \mathrm{i}$.


021832
(1) $0$; (2) $64\mathrm{i}$.


021833
$\dfrac{3}{13}-\dfrac{2}{13}\mathrm{i}$


021834
$5-\dfrac{5}{2}\mathrm{i}$


021835
$x=4,y=3$.


021836
$x=-1,y=5$.


040018
(1) $\dfrac{\pi}{4}$; (2) $\dfrac{\pi}{6}$; (3) $\dfrac{\pi}{10}$; (4) $\dfrac{\pi}{3}$; (5) $\dfrac{5\pi}{12}$; (6) $\dfrac{\pi}{15}$


040019
(1) $60^{\circ}$; (2) $36^{\circ}$; (3) $45^{\circ}$; (4) $75^{\circ}$; (5) $40^{\circ}$; (6) $54^{\circ}$


040020
(1) $2k\pi+\dfrac{\pi}{2}$; (2) $2k\pi+\dfrac{3\pi}{2}$; (3) $2k\pi+\dfrac{7\pi}{6}$; (4) $k\pi+\dfrac{\pi}{4}$; (5) $\dfrac{k\pi}{2}+\dfrac{\pi}{6}$


040021
(1) $k \times 360^{\circ}+60^{\circ}$;\\
(2) $k \times 360^{\circ}+330^{\circ}$; \\
(3) $k \times 360^{\circ}-210^{\circ}$; \\
(4) $k \times 180^{\circ}-45^{\circ}$; \\
(5) $k \times 90^{\circ}+50^{\circ}$


040022
(1) $330^{\circ}$; (2) $240^{\circ}$; (3) $210^{\circ}$; (4) $300^{\circ}$


040023
(1) $\dfrac{4\pi}{3}$; (2) $\dfrac{11\pi}{6}$; (3) $10-2\pi$; (4) $-10+4\pi$


040024
$18$


040025
$3$,$-2$


040026
(1) $1037$; (2) $-4k+53$; (3) $500$


040027
$-2n+10$


040028
15


040029
$7$


040030
$(4,\dfrac{14}{3}]$


040031
$2n-1$


040032
$(3,\dfrac{35}{9})$或$(\dfrac{35}{9},3)$


040033
$200$


040034
略


040035
$a_n=\begin{cases}1, & n=1,\\ 2n, & n=2k, \\ 2n-2, & n=2k+1\end{cases}$($k\in \mathbf{N}$, $k\ge 1$)


040036
$6n-3$


040057
$\dfrac{19}{28}\sqrt{7}$


040058
$\dfrac{79}{156}$


040059
$2$


040060
$-\dfrac{\sqrt{1-m^2}}{m}$


040061
$-\dfrac{1}{5}, \dfrac{1}{5}$


040062
$-\dfrac{1}{3}, 3$


040063
$\dfrac{1}{2}, -2$


040064
$\dfrac{\sqrt{6}}{3}$


040065
$\dfrac{1}{3}, -\dfrac{9}{4}$


040066
$\dfrac{1}{3},  \dfrac{7}{9}$


040067
$\pm\dfrac{\sqrt{2}}{3}$


040068
$\dfrac{1}{4}, \dfrac{2}{5}$


040069
$\dfrac{1-\sqrt{17}}{4}$


040070
(1) 三； (2) 三


040071
(1) $[-\dfrac{1}{2},\dfrac{1}{2})\cup\{1\}$; (2) $[-\dfrac{\pi}{3},\dfrac{\pi}{3})$; (3) $\{-\dfrac{1}{2}\}$


040072
(1) $-\tan \alpha-\cot \alpha$; (2) $-\dfrac{\sqrt{2}}{\sin \alpha}$; (3) $-1$; (4) $0$


040073
略


040074
$-\dfrac{10}{9}$


040075
$a_n=\dfrac{1}{3n-2}$


040076
$a_n=\dfrac{1}{n}$


040077
$(n-\dfrac{4}{5})5^n$


040078
$2^{n+1}-3$


040079
$1078$


040080
$S_n=\begin{cases}\dfrac{n^2}{2}+n-\dfrac 23+\dfrac 23\cdot 2^n, & n\text{为偶数},\\ \dfrac{n^2}{2}-\dfrac 76+\dfrac 23\cdot 2^{n+1}, & n\text{为奇数} \end{cases}$


040081
(1) 略； (2) $n^2$


040082
(1) 不存在； (2) 存在, 如$c_n=2^{n-1}$


040083
$\dfrac{\sqrt{3}}{2}$


040084
$0$


040085
$\{0,-2\pi\}$


040086
$-\dfrac{\pi}6,\dfrac 56\pi$


040087
$\cot \alpha$


040088
$7+4\sqrt{3}$


040089
$\dfrac{\sqrt{2}-\sqrt{6}}{4}$


040090
$\dfrac{\sqrt{3}+\sqrt{35}}{12}$


040091
$\dfrac 12$


040092
$5$


040093
$-\dfrac 12$


040094
$\dfrac{\pi}{12}$


040095
$\{x|x=\pm\frac 23 \pi+2k\pi,k \in \mathbf{Z}\}$


040096
$\dfrac 43 \pi$


040097
\textcircled{4}


040098
C


040099
$\dfrac{-2\sqrt{2}-\sqrt{3}}6$


040100
$-\dfrac 7{25}$


040101
$-\dfrac {\pi}3$


040102
$(-\dfrac {12}{13}, \dfrac{5}{13})$


040103
$(\dfrac {5-12\sqrt{3}}{2}, \dfrac{12-5\sqrt{3}}{2})$


040104
略


040105
$\dfrac {171} {221}, -\dfrac {21} {221}$


040106
$\{-\pi\}$


040107
$\dfrac{8\sqrt{2}-3}{15}$


040108
$\sin \theta$


040109
$-\dfrac{56}{65}$


040110
$\dfrac {\pi}4$


040111
略


040112
略


040181
$\dfrac 7{25}$


040182
$-\dfrac{\pi}3+2k\pi,k \in \mathbf{Z}$


040183
$\dfrac{4\sqrt{3}-3}{10}$


040184
$\dfrac 17$


040185
$4\sqrt{2} \sin(\alpha+\dfrac {7}{4}\pi))$


040186
$3$


040187
$\dfrac 32$


040188
$\sqrt{3}$


040189
$2$


040190
$\dfrac {13}{18}$


040191
$\dfrac{7}{4}\pi$


040192
$\dfrac{64}{25}$


040193
C


040194
A


040195
B


040196
C


040197
$-\dfrac{\pi}6$


040198
$\dfrac 23 \pi$


040199
$\dfrac 32$


040200
$\sqrt{1-k}$


040201
$-\dfrac{484}{729}$


040131
$-\dfrac{25}{12}$


040132
$\dfrac 52$


040133
$-\dfrac{\pi}4$


040134
$-\dfrac 12$


040135
$\dfrac 6{19}$


040136
$-\dfrac {\sqrt{3}}3$


040137
$\dfrac 3{22}$


040138
$4$


040139
$-\dfrac{63}{65}$


040226
$\dfrac 49 \sqrt{2}$


040227
$\sin \theta \cos \theta$


040228
$-\dfrac1{16}$


040229
$\dfrac 32$


040230
$\dfrac{13}{18}$


040231
$-2-\sqrt{7}$


040232
$\sin{\dfrac{\alpha}2}$


040233
$0$


040234
$\dfrac{120}{169}$


040235
$3$或$5$


040236
$\pi-\arcsin{\dfrac{24}{25}}$


040237
$\arcsin{\dfrac{3\sqrt{10}}{10}}$或$\arcsin{\dfrac{\sqrt{10}}{10}}$


040238
$60^{\circ}$或$120^{\circ}$


040239
$\dfrac 23 \pi$


040240
$8$


040241
\textcircled{4}


040242
$\dfrac 35$或$\dfrac{24}{25}$或$\dfrac{3\sqrt{10}}{10}$或$\dfrac{\sqrt{10}}{10}$


040243
(1)$\angle A=75^{\circ}, \angle B=45^{\circ}, a=\sqrt{2}+\sqrt{6}$\\
(2) $\angle B=60^{\circ}, \angle C=75^{\circ}, c=\sqrt{6}+3\sqrt{2}$或
$\angle B=120^{\circ}, \angle C=15^{\circ}, c=3\sqrt{2} - \sqrt{6}$


040244
$\dfrac 12$


040245
$\dfrac 12 \pm \dfrac{\sqrt{6}}5$


040246
$-\dfrac7{25}$


040247
$\dfrac {\sqrt{2}} 2 +\dfrac 14$


040248
$90^\circ$


040249
$\dfrac 1{a}$


040250
$-\dfrac{16}{65}$


040251
$\dfrac{24}{13}$


040252
$\dfrac{\sqrt{11}}{6}$


040253
直角三角形


040254
$120^\circ$


040255
$-\dfrac{48}{49}$


040256
等边三角形


040257
等腰三角形


040258
等腰或直角三角形


040259
$30^\circ$


040260
$30^\circ$或$90^\circ$或$150^\circ$


040261
$2\sqrt{7}$


040262
$\dfrac 12$


040263
$(0,\dfrac{\pi}4]$


040264
(1) $\dfrac 23 \pi$; (2) 等腰钝角三角形


040265
(1) $\dfrac{\sqrt{3}}6$; (2) $\dfrac{\sqrt{39}+\sqrt{3}}2$


040266
$\{x|\dfrac{\pi}6+2k\pi \le x \le \dfrac 56 \pi+2k\pi, k \in  \mathbb{Z} \}$


040267
$[0,3)$


040268
$4$


040269
$\pi$


040270
$\pi$


040271
$\dfrac{\pi}{2}$


040272
$-\sin{\dfrac 12 -1}$


040273
\textcircled{2}\textcircled{3}\textcircled{5}


040274
等腰直角三角形


040275
$\{x|\dfrac{\pi}4+2k\pi \le x \le \dfrac 45 \pi+2k\pi, k \in  \mathbb{Z} \}$


040276
$4\pi$


040277
$\dfrac{\pi}{2}$


040278
$\sqrt{5}$


040279
$12$


040280
$6+\sqrt{15}$


040281
\textcircled{3} \textcircled{4}


040282
$(1)b=1,c=\sqrt{13}$;\\
$(2)$等腰三角形或直角三角形


040396
$\{x|2k\pi+\dfrac{\pi}4<x<2k\pi+\dfrac 34 \pi, k\in \mathbb{Z} \}$


040397
$\{x|2k\pi+\dfrac{\pi}4 \leq x \leq 2k\pi+\dfrac 54 \pi, k\in \mathbb{Z} \}$


040398
$[k\pi+\dfrac{\pi}{4},k\pi+\dfrac 34 \pi],k \in \mathbb{Z}$


040399
$[4k\pi-\dfrac 83 \pi,4k\pi-\dfrac 23 \pi],k \in \mathbb{Z}$


040400
$[2k\pi-\dfrac {\pi}3 ,2k\pi+\dfrac {\pi}6],k \in \mathbb{Z}$


040401
$\{x|x=k\pi-\dfrac 38\pi,k \in \mathbb{Z}\}$


040402
$(-\dfrac 14,2]$


040403
$[-1,2]$


040404
$3,\dfrac{\pi}3$


040405
$y=3\cos(2x+\dfrac{\pi}3)$


040406
$\dfrac 12\sin(2x-\dfrac{\pi}2)+1$


040407
$[\dfrac{197}2 \pi,\dfrac{201}2 \pi)$


040408
$(0,\dfrac 32]$


040409
$[0,\sqrt{2}+\dfrac 32]$


040410
(1)$f(x)=2\sin(\dfrac{\pi}4x+\dfrac{\pi}4)$\\
(2)$x=-\dfrac 23$时，取最大值为$\sqrt{6}$;$x=-4$时，取最小值为$-2\sqrt{2}$


040411
$199$个


040412
(1)$T=\pi$\\
(2)非奇非偶函数\\
(3)增区间为$[k\pi-\dfrac{\pi}3,k\pi+\dfrac{\pi}6],k\in \mathbb{Z}$,减区间为$[k\pi+\dfrac{\pi}{6},k\pi+\dfrac 23\pi],k\in \mathbb{Z}$\\
(4)$y_{min}=1,x=\dfrac{\pi}2;y_{max}=\dfrac 52,x=\dfrac{\pi}6$


040413
$-1-\sqrt{2}<a<1+\sqrt{2}$


040414
$2$


040415
$[2k-1,2k],k \in \mathbb{Z}$


040416
$\dfrac{\pi}2$


040417
$[0,\dfrac 38 \pi]$和$[\dfrac 78 \pi, \pi]$


040418
$\pm \dfrac{\pi}2$


040419
$y=\sin(4x)$


040420
(1)$\dfrac{\pi}6$\\
(2)$\sqrt{3}$


040421
(1)最小正周期为$\pi$，单调减区间为$[k\pi+\dfrac{\pi}{12},k\pi+\dfrac 7{12}\pi],k \in \mathbb{Z}$\\
(2)$y_{\max}=\sqrt{3},x=\dfrac{\pi}6$时取; $y_{\min}=-2,x=\dfrac 7{12}\pi$时取


040527
二


040528
$1$


040529
$2$


040530
$1$


040531
$6$


040532
$\sqrt{2}\pi$


040533
$[\dfrac 12,1)$


040534
$[0,\dfrac 23\pi]$


040535
左，$\dfrac{\pi}6$


040536
$[2k\pi-\dfrac{\pi}3,2k\pi+\dfrac 23 \pi],k \in \mathbb{Z}$


040537
$-1$


040538
B


040539
A


040540
B


040541
\textcircled{2},\textcircled{3},\textcircled{6}


040542
\textcircled{2},\textcircled{3}


040543
\textcircled{1},\textcircled{2},\textcircled{4}


040544
\textcircled{1},\textcircled{2}


040545
\textcircled{1},\textcircled{2},\textcircled{4}


040546
$(\sqrt{3},2\sqrt{7}]$


040547
(1) \textcircled{1} $\varphi=k\pi,k \in \mathbb{Z}$时为奇函数；\\
\textcircled{2} $\varphi=k\pi+\dfrac{\pi}2,k \in \mathbb{Z}$时为偶函数；\\
\textcircled{3} $\varphi \neq \dfrac{k\pi}2,k \in \mathbb{Z}$时为非奇非偶函数.\\
(2)非奇非偶函数


040548
(1)$k=\sqrt{2}$或$k \in [-1,1)$时，一解；$k \in [1,\sqrt{2})$时，两解；$k \in (-\infty，-1)\cup (\sqrt{2},+\infty)$时，无解.
\\
(2)$k=\dfrac 54$或$k \in [-1,1)$时，一解；$k \in [1,\dfrac 54)$时，两解；$k \in (-\infty，-1)\cup (\dfrac 54,+\infty)$时，无解.


040549
(1)不符合；\\
(2)$\theta=\dfrac{\pi}8$时，$S$取最小值，最小值为$12\sqrt{2}-12$


040550
(1)$-\dfrac 12$\\
(2)\\
(3)$\{x|x=k\pi+2\arcsin{\dfrac 16}$或$x=k\pi-\dfrac{\pi}3,k \in \mathbb{Z}\}$


040551
(1)$f(x)=2\sin(2x+\dfrac{\pi}3)$\\
(2)单调增区间为$[k\pi-\dfrac{5\pi}{12},k\pi+\dfrac{\pi}{12}]$,最小值为2，此时$x=k\pi-\dfrac{5\pi}{12}$\\
(3)$0<a\leq \dfrac 4{199\pi}$


040667
$\overrightarrow{0}$


040668
$4$


040669
$\dfrac 92$


040670
$\dfrac 12,\dfrac{\pi}4+k\pi,k \in \mathbb{Z}$


040671
填$(1,2)$内的任意数均可


040672
$-\dfrac 23 \sqrt{3}\overrightarrow{a}$


040673
$[2k\pi-\dfrac{5\pi}6,2k\pi+\dfrac{\pi}6], k \in \mathbb{Z}$


040674
$-\sqrt{2}$


040675
$\{x|-\dfrac{\pi}6+2k\pi<x<\dfrac{\pi}6+2k\pi,k \in \mathbb{Z}\}$


040676
$4$


040677
$[\dfrac{\pi}3,\dfrac{11\pi}3)$


040678
B


040679
A


040680
D


040681
ABD


040682
ACD


040683
(1)$\dfrac 23 \pi$;(2)$6\sqrt{3}$


040684
(1)$\dfrac{\pi}4$;(2)$1$;\\
(3)面积最大值为$4+4\sqrt{2}$,周长的取值范围是$(8,4+4\sqrt{4+2\sqrt{2}})$


040685
(1)最大值为$2$，此时$x=\dfrac{\pi}6+k\pi,k\in\mathbb{Z}$;\\
(2)$[k\pi-\dfrac{\pi}3,k\pi+\dfrac{\pi}6],,k\in\mathbb{Z}$;\\
(3)$y_1=1,y_2=-1,y_1+y_2+y_3+\cdots+y_{2025}=1$


040686
(1)$S=$;\\
(2)$[\sqrt{2}-1,2-\sqrt{2}]$


040687
$\{x|\dfrac{5\pi}{6}+2k\pi<x<\dfrac{13\pi}{6}+2k\pi,k\in\mathbb{Z}\}$


040688
$[1,\sqrt{2}]$


040689
$\sqrt{3}$


040690
$-\dfrac 12 \overrightarrow{a}-\dfrac 12 \overrightarrow{b}$


040691
2,-1


040692
$[-\dfrac{5\pi}8+k\pi,-\dfrac{\pi}8+k\pi],k\in \mathbb{Z}$


040693
$\dfrac 43$


040694
$y=-\sin(2x)$


040695
$(\dfrac{\pi}{12},0),(\dfrac{5\pi}{12},0)$


040696
$-2$


040697
$-\dfrac 32$


040698
$41$


040699
B


040700
D


040701
C


040702
B


040703
$\dfrac{\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c}}3$


040704
$\dfrac 23$


040705
略


040706
(1)$f(\dfrac 12)=\sqrt{2},f(\dfrac 14)=2^{\dfrac 14},f(0)=1$\\
(2)$f(x)=2^{|x|}$\\
(3)周期为2\\
(4)$f(x)=2^{|x-2k|},x \in [2k-1,2k+1],k \in \mathbb{Z}$


040707
充分非必要


040708
$1,\sqrt{7}$


040709
$\dfrac{2\pi}3$


040710
$\pi$


040711
$[2k\pi,2k\pi+\dfrac{\pi}2],k \in \mathbb{Z}$


040712
$\{x|x \neq \dfrac{\pi}4+k\pi,k \in \mathbb{Z}\}$


040713
$[\dfrac{5\pi}6,\dfrac{11\pi}6)$


040714
$-19$


040715
(1)$\omega=2$,严格减区间为$[k\pi+\dfrac{5\pi}{12},k\pi+\dfrac{11\pi}{12}],k \in \mathbb{Z}$\\
(2)$(2,\dfrac{4\sqrt{21}}{3}]$


040716
$(-1,-1)$


040717
$(-1,8),(1,2)$


040718
$4$


040719
$(10,-5)$


040720
$-3,3$


040721
$1,2,-3$


040722
$(2,4),(0,-4),(-2,0)$


040723
$\pm 1$


040724
$-2$


040725
22


040726
B


040727
$-\dfrac{72}{25}$


040728
$14$


040729
$(-\dfrac 79,\dfrac 73)$


040730
A


040731
B


040732
C


040733
$[-37,-13]$


040734
$\dfrac 23$


040735
(1)$\pi-\arccos{\dfrac{\sqrt{21}}{14}}$\\
(2)$k \in (-2,0) \cup (0,+\infty)$


040736
(1)$\overrightarrow{OC}=(\dfrac{1+t}2,-\dfrac{\sqrt{3}}{2} \cdot (1+t)),\overrightarrow{OD}=(\dfrac{2t+1}{2t+2},-\dfrac{\sqrt{3}}{2}  \cdot \dfrac{1}{1+t})$\\
(2)$\dfrac{\pi}3$


040737
(1)$\sqrt{5}$\\(2)$k=-2\pm\sqrt{3}$\\(3)$\theta=\arctan{2}$或$\theta=\pi-\arctan{2}$


040283
$S=T$


040336
$-\dfrac {24}{25}$


040337
$\dfrac{\sqrt{3}}2$


040338
$2$


040339
$2\sin(\alpha+\dfrac 23 \pi)$


040340
$-\dfrac 12$


040341
$-4$


040342
$\{\dfrac {23}{12}\pi,\dfrac{7}{12}\pi\}$


040343
$3$


040344
$-\dfrac 12$


040345
B


040346
C


040347
A


040348
(1)$\dfrac 13$\\
(2)$\dfrac{\pi}4$


040349
(1)$2k$;\\
(2)$10k$;\\
(3)$\dfrac2{25}\sqrt{5}$


040350
$-\dfrac 13$


040351
$x=2k\pi+\dfrac{\pi}2,k \in \mathbb{Z}$


040352
$\pi$


040353
$-\dfrac 45$


040354
$\{x|k\pi \leq x < k\pi+\dfrac{\pi}2,k \in \mathbb{Z}\}$


040355
$-2$


040356
$\{0,\pi,-\dfrac{\pi}3,\dfrac{\pi}3,\dfrac 53 \pi\}$


040357
$(1,+\infty)$


040358
$[2^{-\dfrac 14},4]$


040359
$[0,\dfrac 23\pi]$


040360
$\dfrac {\pi}6$


040361
$16$


040362
D


040363
B


040364
(1)$\omega =2$，定义域为$\{x|x\neq \dfrac{k\pi}2+\dfrac{\pi}8,k \in \mathbb{Z}\}$\\
(2)$\dfrac 43$


040365
(1)$500\sqrt{7}$\\
(2)55706元


040366
(1)$x \leq \log_2 3$\\
(2)$k \in [\dfrac{225}{271},\dfrac{19}9)$


040367
(1)$m=2$\\
(2)在$(-2,2)$上严格减\\
(3)$\dfrac {13}4$


015269
$(0,+\infty)$


015270
$\dfrac{2\pi}{3}$


015271
$-3$


015272
$\dfrac{\pi}{6}$


015273
$3$


015274
$[2k\pi,2k\pi+\pi]$, $k\in \mathbf{Z}$


015275
$f(x)=-1-2x$


015276
$[3,4]$


015277
$1$


015278
\textcircled{1}


015279
$(0,\dfrac{\pi}{4})\cup (\dfrac{3\pi}{4},\pi)$


015280
$[4,+\infty)$


015281
D


015282
A


015283
D


015284
C


015285
$[1,3)$


015286
(1) $\dfrac{3}{5}$; (2) $\dfrac{49}{32}$


015287
(1) $\sqrt{7}$; (2) $\dfrac{9\sqrt{3}}{4}$


015288
(1) $y=3\sin(\dfrac{\pi}{6}x+\dfrac{\pi}{6})+8$, $x\in [0,24]$; (2) 可以进港的时间段为0点至6点, 以及12点至16点; 在0点进港开始卸货, 5点暂时驶离港口, 11点返回港口继续卸货, 16点完成卸货任务


015289
(1) 证明略; (2) $y=F(x)$是$(-\infty,+\infty)$上的严格增函数, 证明略; (3) $y=af(ax+b)$