ĐGTD ĐH Bách khoa - Tư duy Toán học - Phương trình lượng giác thường gặp

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

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

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

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

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

$x=\frac{\pi }{4}+\frac{k\pi }{2}\left(k\in Z\right)$

$x=\frac{3\pi }{4}+2k\pi \left(k\in Z\right)$

$x=\frac{3\pi }{4}+k\pi \left(k\in Z\right)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\left|a\right|>1$

$\left|a\right|=1$

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

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

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

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

$\{\begin{array}{c}x=\frac{\pi }{2}+k2\pi \\ y=\frac{\pi }{6}+k2\pi \end{array}\left(k\in Z\right)$

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

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

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

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

Có 1 họ nghiệm

Có 2 họ nghiệm

Vô nghiệm

Có 1 nghiệm duy nhất

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

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

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

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

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

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

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

Tất cả đều đúng

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

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

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

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

Ba nghiệm

Một nghiệm

Hai nghiệm

Bốn nghiệm

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

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

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

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

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

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

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

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

$x=-\frac{\pi }{6}+\frac{k\pi }{2};x=\frac{\pi }{12}+\frac{k\pi }{2}\left(k\in Z\right)$

$x=\frac{\pi }{12}+\frac{k\pi }{4}\left(k\in Z\right)$

$x=\pm \frac{\pi }{6}+k\pi ;x=-\frac{\pi }{12}+\frac{k\pi }{2}\left(k\in Z\right)$

$x=\frac{\pi }{6}+k\pi ;x=-\frac{\pi }{12}+\frac{k\pi }{2}\left(k\in Z\right)$

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

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

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

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

$x=\frac{\pi }{6}+k\pi ,k\in Z$

$x=-\frac{\pi }{12}+\frac{k\pi }{2},k\in Z$

$x=\frac{2\pi }{3}+k\pi ,k\in Z$

$x=\frac{k\pi }{12};x=\frac{2\pi }{3}+k\pi ,k\in Z$

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

$x=k2\pi ;x=\frac{\pi }{3}+\frac{k2\pi }{3},x=-\frac{\pi }{6}+k2\pi ,x=\frac{7\pi }{6}+k2\pi$

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

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

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

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

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

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

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

$x=\pm \frac{\pi }{3}+\frac{k2\pi }{3}$

$x=\frac{\pi }{2}+k\pi ,x=-\frac{\pi }{3}+\frac{k2\pi }{3}$

$x=2k\pi ,x=\frac{\pi }{2}+\frac{k\pi }{3}$

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9

3

6

7

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

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

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

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

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

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

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

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

Vô nghiệm

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

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

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

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

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

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

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

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1

4

$\frac{24}{5}$

$\frac{28}{5}$

5

$\frac{25}{8}$