Giải các phương trình sau
1. $2\sin 4x+1=0$.
2. $3\cos x-2=0$.
3. $3\tan \left(x+\frac{\pi }{6}\right)-\sqrt{3}=0$.
4. $\cot (3x+1)+1=0$.
5. $(\sin 3x+1)(2\cos x-\sqrt{3})=0$.
6. $\sin 7x+\cos 2x=0$.

(AI giải đề cho tôi)
Bài 3. Giải các phương trình lượng giác.

1) $2\sin 4x+1=0\iff \sin 4x=-\frac{1}{2}$.
$\iff \{\begin{array}{l}4x=-\frac{\pi }{6}+2k\pi \\ 4x=\frac{7\pi }{6}+2k\pi \end{array}\iff \{\begin{array}{l}x=-\frac{\pi }{24}+\frac{k\pi }{2}\\ x=\frac{7\pi }{24}+\frac{k\pi }{2}\end{array}$, $k\in \mathbb{Z}$.

2) $3\cos x-2=0\iff \cos x=\frac{2}{3}$ (thỏa mãn vì $\left|\frac{2}{3}\right|\le 1$).
$\iff x=\pm \arccos \frac{2}{3}+2k\pi$, $k\in \mathbb{Z}$.

3) $3\tan \left(x+\frac{\pi }{6}\right)-\sqrt{3}=0\iff \tan \left(x+\frac{\pi }{6}\right)=\frac{\sqrt{3}}{3}$ (điều kiện: $x+\frac{\pi }{6}\ne \frac{\pi }{2}+k\pi$).
$\iff x+\frac{\pi }{6}=\frac{\pi }{6}+k\pi \iff x=k\pi$, $k\in \mathbb{Z}$. (Các nghiệm này thỏa mãn điều kiện.)

4) $\cot (3x+1)+1=0\iff \cot (3x+1)=-1$ (điều kiện: $3x+1\ne k\pi$).
$\iff 3x+1=-\frac{\pi }{4}+k\pi \iff x=-\frac{\pi }{12}-\frac{1}{3}+\frac{k\pi }{3}$, $k\in \mathbb{Z}$.

5) $(\sin 3x+1)(2\cos x-\sqrt{3})=0\iff \sin 3x=-1$ hoặc $\cos x=\frac{\sqrt{3}}{2}$.
• $\sin 3x=-1\iff 3x=-\frac{\pi }{2}+2k\pi \iff x=-\frac{\pi }{6}+\frac{2k\pi }{3}$, $k\in \mathbb{Z}$.
• $\cos x=\frac{\sqrt{3}}{2}\iff x=\pm \frac{\pi }{6}+2k\pi$, $k\in \mathbb{Z}$.

6) $\sin 7x+\cos 2x=0\iff \sin 7x=-\cos 2x=\sin \left(2x-\frac{\pi }{2}\right)$.
$\iff \{\begin{array}{l}7x=2x-\frac{\pi }{2}+2k\pi \\ 7x=\pi -\left(2x-\frac{\pi }{2}\right)+2k\pi \end{array}$
$\iff \{\begin{array}{l}x=-\frac{\pi }{10}+\frac{2k\pi }{5}\\ 9x=\frac{3\pi }{2}+2k\pi \iff x=\frac{\pi }{6}+\frac{2k\pi }{9}\end{array}$, $k\in \mathbb{Z}$.

Vậy tập nghiệm lần lượt là:
1) $S=\left\{-\frac{\pi }{24}+\frac{k\pi }{2};\frac{7\pi }{24}+\frac{k\pi }{2}\right\}$, $k\in \mathbb{Z}$.
2) $S=\left\{\pm \arccos \frac{2}{3}+2k\pi \right\}$, $k\in \mathbb{Z}$.
3) $S=\{k\pi \}$, $k\in \mathbb{Z}$.
4) $S=\left\{-\frac{\pi }{12}-\frac{1}{3}+\frac{k\pi }{3}\right\}$, $k\in \mathbb{Z}$.
5) $S=\left\{-\frac{\pi }{6}+\frac{2k\pi }{3};\pm \frac{\pi }{6}+2k\pi \right\}$, $k\in \mathbb{Z}$.
6) $S=\left\{-\frac{\pi }{10}+\frac{2k\pi }{5};\frac{\pi }{6}+\frac{2k\pi }{9}\right\}$, $k\in \mathbb{Z}$.