ans

018366
(1) $x=\dfrac{\pi}{3}+2k\pi$或$\dfrac{2\pi}{3}+2k\pi$, $k\in \mathbf{Z}$; (2) $x=\pm \dfrac{3\pi}{4}+2k\pi$, $k\in \mathbf{Z}$; (3) $x=k\pi+\dfrac{\pi}{6}$, $k\in \mathbf{Z}$

018367
(1) $\{\dfrac{\pi}{6},\dfrac{\pi}{3},\dfrac{7\pi}{6},\dfrac{4\pi}{3}\}$; (2) $\{x|x=2k\pi+\dfrac{\pi}{12}\text{或}x=2k\pi-\dfrac{5\pi}{12}, \ k\in \mathbf{Z}\}$; (3) $\{x|x=\dfrac{k\pi}{2}+\dfrac{\pi}{4}, \ k\in \mathbf{Z}\}$

018369
(1) $x=-\dfrac{\pi}{2}+2k\pi$, $k\in \mathbf{Z}$; (2) $x=\pm \dfrac{\pi}{6}+2k\pi$, $k\in \mathbf{Z}$; (3) $x=\dfrac{\pi}{4}+k\pi$, $k\in \mathbf{Z}$

009563
(1) $\{\dfrac{\pi}{2},\dfrac{11\pi}{6}\}$; (2) $\{x|x=k\pi+\dfrac{5\pi}{24}\text{或}k\pi-\dfrac{11\pi}{24}, \ k\in \mathbf{Z}\}$; (3) $\{x|x=\dfrac{k\pi}{3}+\dfrac{\pi}{6}, \ k\in \mathbf{Z}\}$

018372
$\dfrac{\sqrt{6}-\sqrt{2}}{4}$, $\dfrac{\sqrt{6}+\sqrt{2}}{4}$

018373
$-\dfrac{56}{65}$

018375
$\dfrac{\pi}{3}$

009564
(1) $\dfrac{\sqrt{2}}{2}$; (2) $\dfrac{\sqrt{3}}{2}$

009565
$-\dfrac{7\sqrt{2}}{26}$

018377
(1) $\dfrac{\sqrt{3}}{2}$; (2) $\cos 2\alpha$; (3) $-\cos 3x$

018376
$-\dfrac{4}{5}$

018378
$\dfrac{\sqrt{6}-\sqrt{2}}{4}$

018379
证明略

018380
(1) $-1$; (2) $\dfrac{1}{7}$

018381
$-\sqrt{3}$

009567
(1) $\dfrac{\sqrt{3}}{2}$; (2) $\sqrt{3}$

009568
$\dfrac{\sqrt{2}}{10}$, $7$

009569
(1) 证明略; (2) 证明略

018385
证明略

018386
$(-\dfrac{\sqrt{2}}{2},\dfrac{3\sqrt{2}}{2})$

018387
(1) $\sin(\alpha+\dfrac{\pi}{3})$; (2) $\sqrt{2}\sin(\alpha-\dfrac{\pi}{4})$; (3) $5\sin(\alpha+\varphi)$, 其中$\cos\varphi=\dfrac{3}{5}$, $\sin\varphi=\dfrac{4}{5}$; (4) $\sqrt{a^2+b^2}\sin(\alpha+\varphi)$, 其中$\cos\varphi=\dfrac{a}{\sqrt{a^2+b^2}}$, $\sin\varphi=\dfrac{b}{\sqrt{a^2+b^2}}$

024614
$\{\alpha|\alpha=2k\pi\text{或}\dfrac{2\pi}{3}+2k\pi, \ k\in \mathbf{Z}\}$

009570
$\sin C=\dfrac{220}{221}$, $\cos C=-\dfrac{21}{221}$

009571
$\sin(\alpha+\beta)=\dfrac{33}{65}$, $\cos(\alpha+\beta)=-\dfrac{56}{65}$, $\alpha+\beta$是第二象限角

009572
(1) $\sqrt{2}\sin(\alpha+\dfrac{\pi}{4})$; (2) $2\sin(\alpha+\dfrac{2\pi}{3})$

018390
$\dfrac{\sqrt{6}-\sqrt{2}}{4}$

018393
$\sin 2\alpha=-\dfrac{24}{25}$, $\cos 2\alpha = -\dfrac{7}{25}$, $\tan 2\alpha = \dfrac{24}{7}$

018395
$\cos 3\theta = 4\cos^3\theta-3\cos \theta$

018396
(1) 证明略; (2) 证明略

009573
(1) $\dfrac{1}{4}$; (2) $\dfrac{\sqrt{2}}{2}$; (3) $\dfrac{\sqrt{3}}{6}$

018394
$\sin 4\alpha=\dfrac{336}{625}$; $\cos 4\alpha = -\dfrac{527}{625}$

009574
$\sin 2\alpha=-\dfrac{4}{5}$, $\cos 2\alpha=-\dfrac{3}{5}$, $\tan 2\alpha=\dfrac{4}{3}$

009575
(1) 证明略; (2) 证明略; (3) 证明略

018398
$\dfrac{7}{25}$

018401
$\cos ^2 \dfrac{\alpha}{2}=\dfrac{1+\cos\alpha}{2}$, $\sin ^2 \dfrac{\alpha}{2}=\dfrac{1-\cos\alpha}{2}$, $\tan ^2 \dfrac{\alpha}{2}=\dfrac{1-\cos\alpha}{1+\cos\alpha}$

018402
证明略

018403
证明略

018404
证明略

009576
证明略

009577
证明略

009578
证明略

018405
证明略

040387
$\dfrac{5\pi}{3}$

040388
$-1$

040389
$-\dfrac{\pi}{3}$或$\dfrac{4\pi}{3}$

040390
$-3$

040391
$-2$

040392
$-\dfrac{\sqrt{23}}{4}$