giúp vs,e sẽ tick sau
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Câu 49. Phương trình \(2 \cot x - \sqrt{3} = 0\) có nghiệm là
A. \(x = \frac{\pi}{6} + k2\pi \, (k \in \mathbb{Z})\).
B. \(x = \frac{\pi}{3} + k2\pi \, (k \in \mathbb{Z})\).
C. \(x = \text{arccot} \frac{\sqrt{3}}{2} + k2\pi \, (k \in \mathbb{Z})\).
D. \(x = -\frac{\pi}{6} + k\pi \, (k \in \mathbb{Z})\).
Câu 50. Giải phương trình \(\cot (3x - 1) = \sqrt{3}\).
A. \(x = \frac{1}{3} + \frac{5\pi}{18} + k\pi \, (k \in \mathbb{Z})\).
B. \(x = \frac{1}{3} + \frac{\pi}{18} + \frac{\pi}{3} + k\pi \, (k \in \mathbb{Z})\).
C. \(x = \frac{5\pi}{18} + k\pi \, (k \in \mathbb{Z})\).
D. \(x = \frac{1}{3} + \frac{\pi}{3} + k\pi \, (k \in \mathbb{Z})\).
Câu 51. Phương trình \(\sin 2x = \cos x\) có nghiệm là
A.
\[
x = \frac{\pi}{6} + \frac{k\pi}{3} \, (k \in \mathbb{Z}).
\]
\(x = \frac{\pi}{2} + k2\pi \, (k \in \mathbb{Z})\).
B.
\[
x = \frac{\pi}{6} + \frac{k\pi}{3} \, (k \in \mathbb{Z}).
\]
\(x = \frac{\pi}{3} + k2\pi \, (k \in \mathbb
Câu 49: \(2\cdot\cot x-\sqrt3=0\)
=>\(2\cdot\cot x=\sqrt3\)
=>\(\cot x=\frac{\sqrt3}{2}\)
=>\(x=arc\cot\left(\frac{\sqrt3}{2}\right)+k\pi\)
=>Chọn C
Câu 50: \(\cot\left(3x-1\right)=-\sqrt3\)
=>\(3x-1=-\frac{\pi}{6}+k\pi\)
=>\(3x=-\frac{\pi}{6}+k\pi+1\)
=>\(x=-\frac{\pi}{18}+\frac{k\pi}{3}+\frac13=-\frac{\pi}{18}+\frac{\pi}{3}+\frac{\left(k-1\right)\cdot\pi}{3}+\frac13=\frac{5}{18}\pi+\left(k-1\right)\cdot\frac{\pi}{3}+\frac13\)
=>Chọn A
Câu 51: sin 2x =cosx
=>\(\sin2x=\sin\left(\frac{\pi}{2}-x\right)\)
=>\(\left[\begin{array}{l}2x=\frac{\pi}{2}-x+k2\pi\\ 2x=\pi-\frac{\pi}{2}+x+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}3x=\frac{\pi}{2}+k2\pi\\ x=\frac{\pi}{2}+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{\pi}{6}+\frac{k2\pi}{3}\\ x=\frac{\pi}{2}+k2\pi\end{array}\right.\)
=>Chọn D
Câu 52: sin 3x=cosx
=>\(\sin3x=\sin\left(\frac{\pi}{2}-x\right)\)
=>\(\left[\begin{array}{l}3x=\frac{\pi}{2}-x+k2\pi\\ 3x=\pi-\frac{\pi}{2}+x+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}4x=\frac{\pi}{2}+k2\pi\\ 2x=\frac{\pi}{2}+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{\pi}{8}+\frac{k\pi}{2}\\ x=\frac{\pi}{4}+k\pi\end{array}\right.\)
=>Chọn B
Câu 53: \(\sin\left(3x+\frac{\pi}{3}\right)=-\frac{\sqrt3}{2}\)
=>\(\left[\begin{array}{l}3x+\frac{\pi}{3}=-\frac{\pi}{3}+k2\pi\\ 3x+\frac{\pi}{3}=\pi+\frac{\pi}{3}+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}3x=-\frac23\pi+k2\pi\\ 3x=\pi+k2\pi\end{array}\right.\)
=>\(\left[\begin{array}{l}x=-\frac29\pi+\frac{k2\pi}{3}\\ x=\frac{\pi}{3}+\frac{k2\pi}{3}\end{array}\right.\)
x∈\(\left(0;\frac{\pi}{2}\right)\)
=>\(\left[\begin{array}{l}-\frac29\pi+\frac{k2\pi}{3}\in\left(0;\frac{\pi}{2}\right)\\ \frac{\pi}{3}+\frac{k2\pi}{3}\in\left(0;\frac{\pi}{2}\right)\end{array}\right.\Rightarrow\left[\begin{array}{l}-\frac29+\frac{2k}{3}\in\left(0;\frac12\right)\\ \frac13+\frac{2k}{3}\in\left(0;\frac12\right)\end{array}\right.\)
=>\(\left[\begin{array}{l}\frac{2k}{3}\in\left(\frac29;\frac{13}{18}\right)\\ \frac{2k}{3}\in\left(-\frac13;\frac16\right)\end{array}\right.\Rightarrow\left[\begin{array}{l}k\in\left(\frac13;\frac{13}{12}\right)\\ k\in\left(-\frac12;\frac14\right)\end{array}\right.\Rightarrow\left[\begin{array}{l}k=1\\ k=0\end{array}\right.\)
=>Có 2 nghiệm
=>Chọn D
