ĐGNL ĐHQG Hà Nội - Tư duy định lượng - Phương trình lượng giác thường gặp

$[\begin{array}{c}x=\frac{k\pi }{2}\\ x=\pm \frac{1}{2}arccos(-\frac{1}{6})+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{k\pi }{2}\\ x=\pm \frac{5}{2}arccos(-\frac{1}{6})+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{k\pi }{2}\\ x=\pm \frac{1}{2}arccos(-\frac{1}{3})+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{k\pi }{2}\\ x=\pm \frac{1}{3}arccos(-\frac{1}{6})+k\pi \end{array}(k\in Z)$

$$\left|a\right|\ge 1$$

$$\left|a\right|>1$$

$$\left|a\right|=1$$

$$\left|a\right|\ne 1$$

$\{\begin{array}{c}x=\frac{\pi }{6}+k2\pi \\ y=-\frac{\pi }{6}+k2\pi \end{array}(k\in Z)$

$\{\begin{array}{c}x=\frac{2\pi }{3}+k2\pi \\ y=\frac{\pi }{3}-k2\pi \end{array}(k\in Z)$

$\{\begin{array}{c}x=\frac{2\pi }{3}+k2\pi \\ y=\frac{\pi }{3}+k2\pi \end{array}(k\in Z)$

$\{\begin{array}{c}x=\frac{\pi }{6}+k2\pi \\ y=\frac{\pi }{6}+k2\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{\pi }{3}+k\pi \\ x=\frac{\pi }{6}+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{\pi }{3}+k2\pi \\ x=\frac{\pi }{6}+k2\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=-\frac{\pi }{3}+k\pi \\ x=-\frac{\pi }{6}+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=-\frac{\pi }{3}+k2\pi \\ x=\frac{\pi }{6}+k\pi \end{array}(k\in Z)$

$$x=-\frac{\pi }{6}+k2\pi \left(k\in Z\right)$$

$$x=\frac{\pi }{6}+\frac{k2\pi }{3}\left(k\in Z\right)$$

$$x=-\frac{\pi }{6}+\frac{k2\pi }{3}\left(k\in Z\right)$$

$$x=\pm \frac{\pi }{6}+\frac{k2\pi }{3}\left(k\in Z\right)$$

$$x=\pm \frac{\pi }{6}+k\pi \left(k\in Z\right)$$

$$x=\pm \frac{\pi }{6}+k2\pi \left(k\in Z\right)$$

$$x=\pm \frac{\pi }{3}+k\pi \left(k\in Z\right)$$

$[\begin{array}{c}x=\frac{\pi }{6}+k\pi \left(k\in Z\right)\\ x=\frac{\pi }{3}+k\pi \left(k\in Z\right)\end{array}$

0

1

2

4

m=1

Không có m

m=0

Với mọi m

$[\begin{array}{c}x=k\pi \\ x=\frac{\pi }{3}+k\pi \end{array}(k\in \mathbb{Z})$

$[\begin{array}{c}x=k\pi \\ x=\frac{2\pi }{3}+k2\pi \end{array}(k\in \mathbb{Z})$

$[\begin{array}{c}x=k2\pi \\ x=\frac{2\pi }{3}+k2\pi \end{array}(k\in \mathbb{Z})$

$[\begin{array}{c}x=k\pi \\ x=\frac{2\pi }{3}+k\pi \end{array}(k\in \mathbb{Z})$

Có 1 họ nghiệm

Có 2 họ nghiệm

Vô nghiệm

Có 1 nghiệm duy nhất

$$-\frac{5{\pi }^2}{12}$$

$$-\frac{5{\pi }^2}{144}$$

$$\frac{5{\pi }^2}{144}$$

$$\frac{{\pi }^2}{12}$$

0

1

2

3

$$\frac{3\pi }{5}$$

$$\frac{29\pi }{30}$$

$$\frac{5\pi }{6}$$

$$\frac{23\pi }{30}$$

$$x=k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{2}+k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{6}+k2\pi \left(k\in \mathbb{Z}\right)$$

Tất cả đều đúng.

$[\begin{array}{c}x=\frac{\pi }{2}+k\pi \\ x=\frac{\pi }{6}+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{\pi }{4}+k\pi \\ x=\frac{\pi }{3}+k\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{\pi }{8}+k2\pi \\ x=\frac{\pi }{12}+k2\pi \end{array}(k\in Z)$

$[\begin{array}{c}x=\frac{\pi }{8}+k\pi \\ x=\frac{\pi }{12}+k\pi \end{array}(k\in Z)$

Ba nghiệm

Một nghiệm

Hai nghiệm

Bốn nghiệm

0

1

2

3

9

3

6

7

$$m\ne \frac{1}{2}$$

$$m=\frac{1}{2}$$

$\{\begin{array}{c}\frac{1}{3}<m<1\\ m\ne \frac{1}{2}\end{array}$

</>

$$\frac{1}{3}<m<1$$

</>

$$x=\frac{\pi }{9}+\frac{k2\pi }{3};\left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{18}+\frac{k\pi }{6};\left(k\in \mathbb{Z}\right)$$

$$x=\pm \frac{\pi }{6}+\frac{k\pi }{2}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{18}+\frac{k\pi }{3};x=-\frac{\pi }{6}+\frac{k\pi }{2}\left(k\in \mathbb{Z}\right)$$

$$x=-\frac{\pi }{4}+k\pi ;x=\frac{\pi }{6}+k2\pi ;x=\frac{5\pi }{6}+k2\pi ;x=-\frac{\pi }{2}+k2\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{4}+k2\pi ;x=-\frac{\pi }{6}+k2\pi ;x=\frac{5\pi }{6}+k\pi ;x=-\frac{\pi }{2}+k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\pm \frac{\pi }{6}+k2\pi ;x=\frac{5\pi }{6}+k2\pi ;x=-\frac{\pi }{2}+k2\pi \left(k\in \mathbb{Z}\right)$$

$$x=-\frac{\pi }{8}+k\pi ;x=\frac{\pi }{6}+k\pi ;x=-\frac{5\pi }{6}+\frac{k\pi }{6};x=-\frac{\pi }{2}+\frac{k\pi }{6}\left(k\in \mathbb{Z}\right)$$

$$x=k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{2\pi }{3}+2k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{3}+2k\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{k\pi }{3}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{n\pi }{2};x=\frac{\pi }{20}+\frac{k\pi }{13}\left(k,n\in \mathbb{Z}\right)$$

$$x=n\pi ;x=\frac{\pi }{20}+\frac{k\pi }{10}\left(k,n\in \mathbb{Z}\right)$$

$$x=n\pi ;x=\frac{3\pi }{5}+\frac{2k\pi }{7}\left(k,n\in \mathbb{Z}\right)$$

$$x=n\pi ;x=\frac{3\pi }{5}+\frac{7k\pi }{13}\left(k,n\in \mathbb{Z}\right)$$

$$x=\pm \frac{\pi }{3}+k2\pi ;x=\frac{2\pi }{3}+k2\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{4}+k\pi ;x=\frac{\pi }{6}+k\pi \left(k\in \mathbb{Z}\right)$$

$$x=k\pi ;x=\frac{\pi }{3}+k2\pi ;x=\frac{2\pi }{3}+k2\pi \left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{2}+k\pi ;x=\frac{\pi }{6}+\frac{k\pi }{3}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{k\pi }{18};x=\frac{k\pi }{22}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{k\pi }{9};x=\frac{\pi }{44}+\frac{k\pi }{22}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{\pi }{3}+\frac{k\pi }{18};x=\frac{\pi }{22}+\frac{k\pi }{22}\left(k\in \mathbb{Z}\right)$$

$$x=\frac{k\pi }{3};x=\frac{\pi }{44}+\frac{k\pi }{44}\left(k\in \mathbb{Z}\right)$$

$$x=k\pi ,x=\frac{\pi }{6}+\frac{k\pi }{3},x=\frac{\pi }{12}+k\pi ,x=\frac{5\pi }{7}+k\pi$$

$$x=k2\pi ,x=\frac{7\pi }{6}+k2\pi ,x=-\frac{\pi }{6}+k2\pi ,x=\frac{7\pi }{6}+k2\pi$$

$$x=k\pi ,x=\frac{\pi }{3}+\frac{k2\pi }{3},x=\frac{\pi }{12}+k\pi ,x=\frac{5\pi }{12}+k\pi$$

$$x=k2\pi ,x=\frac{\pi }{6}+\frac{k2\pi }{3},x=\frac{\pi }{12}+k\pi ,x=\frac{7\pi }{12}+k\pi$$

$$x=\frac{\pi }{2}+k\pi ,x=\pm \frac{1}{5}\arccos \frac{1+\sqrt{17}}{8}+k\pi ,x=\pm \frac{1}{5}\arccos \frac{1-\sqrt{17}}{8}+k\pi$$

$$x=\pm \frac{\pi }{6}+k\pi$$

$$x=\pm \frac{1}{2}\arccos \frac{1+\sqrt{15}}{7}+k\pi ,x=\pm \frac{1}{2}\arccos \frac{1-\sqrt{15}}{7}+k\pi$$

$$x=\frac{\pi }{2}+k\pi ,x=\pm \frac{1}{2}\arccos \frac{1+\sqrt{17}}{8}+k\pi ,x=\pm \frac{1}{2}\arccos \frac{1-\sqrt{17}}{8}+k\pi$$