4. Giải các phương trình: a) sin(x+1)=$\frac{2}{3}$;            b)${\sin }^{2}2x=\frac{1}{2}$

                                        c)${\cos }^2\frac{x}{2}=\frac{1}{3}$              d) tan$\left(\frac{\mathrm{\pi }}{12}+12x\right)=-\sqrt{3}$

Giải:

a) sin(x+1)=$\frac{2}{3}$$\iff {[}_{x+1=\mathrm{\pi }-\arcsin \frac{2}{3}+\mathrm{k}2\mathrm{\pi }}^{x+1=arc\sin \frac{2}{3}+k2\mathrm{\pi }}$

                         $\iff {[}_{x=-1+\mathrm{\pi }-\arcsin \frac{2}{3}+\mathrm{k}2\mathrm{\pi }}^{x=-1+arc\sin \frac{2}{3}+k2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}})$

b) ${\sin }^{2}2x = \frac{1}{2}\iff \frac{1-\cos 4x}{2}=\frac{1}{2}\iff \cos 4x=0$

                        $\iff 4x=\frac{\mathrm{\pi }}{2}+k\mathrm{\pi }\iff \mathrm{x}=\frac{\mathrm{\pi }}{8}+\mathrm{k}\frac{\mathrm{\pi }}{4}; \mathrm{k}\in \mathrm{\mathbb{Z}}$

c) $co{t}^2\frac{x}{2}=\frac{1}{3}\iff cot\frac{x}{2}=\pm \frac{\sqrt{3}}{3}\iff \frac{x}{2}=\pm \frac{\mathrm{\pi }}{3}+k\mathrm{\pi }\iff \mathrm{x}=\pm \frac{2\mathrm{\pi }}{3}+\mathrm{k}2\mathrm{\pi }, \mathrm{k}\in \mathrm{\mathbb{Z}}$

d) $\tan \left(\frac{\mathrm{\pi }}{12}+12x\right)=-\sqrt{3}\iff \tan \left(\frac{\mathrm{\pi }}{12}+12x\right)=\tan \left(-\frac{\mathrm{\pi }}{3}\right)$

$\iff \frac{\mathrm{\pi }}{12}+12x=-\frac{\mathrm{\pi }}{3}+k\mathrm{\pi }\iff 12x =-\frac{5\mathrm{\pi }}{12}+\mathrm{k\pi }\iff \mathrm{x}=-\frac{5\mathrm{\pi }}{144}+\mathrm{k}\frac{\mathrm{\pi }}{12}, \mathrm{k}\in \mathrm{\mathbb{Z}}$