- Phương trình lượng giác đưa về dạng tích (có lời giải chi tiết)

Câu 1 : Giải phương trình: \(1 + \cos x + \cos 2x + \cos 3x = 0\)

A \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = \frac{\pi }{3} + \frac{{2m\pi }}{3}\end{array} \right.\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

B \(\left[ \begin{array}{l}x = \frac{\pi }{2} + \frac{{k\pi }}{2}\\x = \frac{\pi }{3} + \frac{{2m\pi }}{3}\end{array} \right.\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

C \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k\pi \\x = \frac{\pi }{3} + \frac{{2m\pi }}{3}\end{array} \right.\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

D \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k\pi \\x = \frac{\pi }{3} + \frac{{m\pi }}{3}\end{array} \right.\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

Câu 2 : Cho phương trình: \({\sin ^2}x + {\sin ^2}2x + {\sin ^2}3x = \frac{3}{2}\), nghiệm của pt là:

A \(\left[ \begin{array}{l}x = \frac{\pi }{8} + \frac{{k\pi }}{2}\\x = \frac{\pi }{3} + m2\pi \\x = - \frac{\pi }{3} + l2\pi \end{array} \right.\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

B \(\left[ \begin{array}{l}x = \frac{\pi }{8} + \frac{{k\pi }}{4}\\x = \frac{\pi }{3} + m\pi \\x = - \frac{\pi }{3} + l\pi \end{array} \right.\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

C \(\left[ \begin{array}{l}x = \frac{\pi }{8} + \frac{{k\pi }}{4}\\x = \frac{\pi }{3} + m2\pi \\x = - \frac{\pi }{3} + l2\pi \end{array} \right.\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

D \(\left[ \begin{array}{l}x = \frac{\pi }{8} + \frac{{k\pi }}{2}\\x = \frac{\pi }{3} + m\pi \\x = - \frac{\pi }{3} + l\pi \end{array} \right.\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

Câu 3 : Cho phương trình: \(\sin x - \sin 2x + \sin 3x = 0\), nghiệm của pt là:

A \(x = k\frac{\pi }{2} + k\pi ,k \in \mathbb{Z}\)

B \(x = \pm \frac{\pi }{3} + k2\pi ;x = k\frac{\pi }{2},k \in \mathbb{Z}\)

C \(x = \pm \frac{\pi }{6} + k\pi ,k \in \mathbb{Z}\)

D \(x = \pm \frac{\pi }{3} + k\pi ,k \in \mathbb{Z}\)

Câu 4 : Giải phương trình : \(2{\cos ^3}x + \cos 2x + \sin x = 0\)

A \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = - \frac{\pi }{4} + m2\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

B \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k\pi \\x = - \frac{\pi }{4} + m\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

C \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k\pi \\x = - \frac{\pi }{4} + m2\pi \end{array} \right.\,\,\,\,\,\,\left( {\,k,\;m \in \mathbb{Z}} \right)\)

D \(\left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = - \frac{\pi }{4} + m\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

Câu 5 : Tập nghiệm của phương trình \(2{\sin ^3}x - \cos 2x + \cos x = 0\) là?

A \(\left[ \begin{array}{l}x = k2\pi \\x = - \frac{\pi }{4} + m\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

B \(\left[ \begin{array}{l}x = k\pi \\x = - \frac{\pi }{4} + m\pi \end{array} \right.\,\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

C \(\left[ \begin{array}{l}x = k2\pi \\x = - \frac{\pi }{4} + m2\pi \end{array} \right.\,\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

D \(\left[ \begin{array}{l}x = k\pi \\x = - \frac{\pi }{4} + m2\pi \end{array} \right.\,\,\,\,\,\,\,\,\left( {k,\;m \in \mathbb{Z}} \right)\)

Câu 6 : Giải phương trình: \({\cos ^4}\frac{x}{2} - {\sin ^4}\frac{x}{2} = \sin 2x\,\,\,\,\,\,\)

A \(\left[ \begin{array}{l}x = \frac{\pi }{6} + k2\pi \\x = \frac{{5\pi }}{6} + m2\pi \\x = \frac{\pi }{2} + l\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

B \(\left[ \begin{array}{l}x = \frac{\pi }{6} + k2\pi \\x = \frac{{5\pi }}{6} + m2\pi \\x = \frac{\pi }{2} + l2\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

C \(\left[ \begin{array}{l}x = \frac{\pi }{6} + k\pi \\x = \frac{{5\pi }}{6} + m\pi \\x = \frac{\pi }{2} + l2\pi \end{array} \right.\,\,\,\,\,\,\,\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

D \(\left[ \begin{array}{l}x = \frac{\pi }{6} + k\pi \\x = \frac{{5\pi }}{6} + m\pi \\x = \frac{\pi }{2} + l\pi \end{array} \right.\,\,\,\,\,\,\left( {k,\;m,\;l \in \mathbb{Z}} \right)\)

Câu 7 : Phương trình \(\cos 3x + \cos 2x - \cos x - 1 = 0\) có tập nghiệm là

A \(\left[ \begin{array}{l}x = k\pi \\x = \pm \frac{{2\pi }}{3} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right).\)

B \(\left[ \begin{array}{l}x = k2\pi \\x = \pm \frac{{2\pi }}{3} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right).\)

C \(\left[ \begin{array}{l}x = k2\pi \\x = \frac{{2\pi }}{3} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right).\)

D \(\left[ \begin{array}{l}x = k\pi \\x = \pm \frac{{2\pi }}{3} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right).\)

Câu 8 : Phương trình: \(4.\frac{{{{\sin }^6}x + {{\cos }^6}x}}{{\sin 2x}} = \frac{1}{2}\left( {{{\tan }^2}x + {{\cot }^2}x} \right)\) có bao nhiêu nghiệm thuộc \(\left[ {0;\frac{\pi }{2}} \right]\)

A 3

B 2

C 4

D 5

Câu 9 : Phương trình: \({\sin ^7}x + {\cos ^5}x + \frac{1}{2}({\sin ^5}x + {\cos ^3}x)\sin 2x = \sin x + \cos x\) có bao nhiêu nghiệm thuộc \(\left[ {0;10\pi } \right]\)

A 20