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Câu 1: \(\tan x=\tan\left(\frac{6\pi}{5}\right)\)
=>\(x=\frac{6\pi}{5}+k\pi\)
=>Nghiệm nguyên dương nhỏ nhất là \(\frac{6\pi}{5}-\pi=\frac15\pi\)
=>Chọn A
Câu 2: \(\cot2x=\cot\left(\frac{\pi}{2}-x\right)\)
=>\(2x=\frac{\pi}{2}-x+k\pi\)
=>\(3x=\frac{\pi}{2}+k\pi\)
=>\(x=\frac{\pi}{6}+\frac{k\pi}{3}\)
mà \(x\in\left\lbrack0;\pi\right\rbrack\)
nên \(x\in\left\lbrace\frac{\pi}{6};\frac{\pi}{2};\frac56\pi\right\rbrace\)
=>Chọn B
Câu 3:
\(4\cdot sin^22x-1=0\)
=>\(4\cdot sin^22x=1\)
=>\(\sin^22x=\frac14\)
=>\(\left[\begin{array}{l}\sin2x=\frac12\\ \sin2x=-\frac12\end{array}\right.\)
TH1: sin 2x=1/2
=>\(\left[\begin{array}{l}2x=\frac{\pi}{6}+k2\pi\\ 2x=\pi-\frac{\pi}{6}+k2\pi=\frac56\pi+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{\pi}{12}+k\pi\\ x=\frac{5}{12}\pi+k\pi\end{array}\right.\)
mà \(x\in\left(-\frac{\pi}{2};\frac{\pi}{2}\right)\)
nên \(x\in\left(\frac{\pi}{12};\frac{5}{12}\pi;-\frac{1}{12}\pi\right)\)
TH2: sin 2x=-1/2
=>\(\left[\begin{array}{l}2x=\frac{-\pi}{6}+k2\pi\\ 2x=\pi-\frac{-\pi}{6}+k2\pi=\frac76\pi+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-\frac{\pi}{12}+k\pi\\ x=\frac{7}{12}\pi+k\pi\end{array}\right.\)
mà \(x\in\left(-\frac{\pi}{2};\frac{\pi}{2}\right)\)
nên \(x\in\left(-\frac{\pi}{12};-\frac{5}{12}\pi\right)\)
Tổng các nghiệm là \(\frac{\pi}{12}+\frac{5\pi}{12}-\frac{1}{12}\pi-\frac{\pi}{12}-\frac{5}{12}\pi=-\frac{1}{12}\pi\)
Câu 4: \(cos\left(x+\frac{\pi}{4}\right)=\frac12\)
=>\(\left[\begin{array}{l}x+\frac{\pi}{4}=\frac{\pi}{3}+k2\pi\\ x+\frac{\pi}{4}=-\frac{\pi}{3}+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{\pi}{12}+k2\pi\\ x=-\frac{7}{12}\pi+k2\pi\end{array}\right.\)
mà \(x\in\left(-\pi;\pi\right)\)
nên \(x\in\left(\frac{\pi}{12};-\frac{7}{12}\pi\right)\)
=>Tổng các nghiệm là:
\(\frac{\pi}{12}-\frac{7}{12}\pi=-\frac{6}{12}\pi=-\frac12\pi\)
=>Chọn B
D=sin(pi+x)+sinx+cot(pi-x)+tan(pi/2-x)
=-sinx+sinx-cotx+cotx=0
\(1-2cos^2x-sinx=0\)
\(\Leftrightarrow1-2\left(1-sin^2x\right)-sinx=0\)
\(\Leftrightarrow2sin^2x-sinx-1=0\Rightarrow\left[{}\begin{matrix}sinx=1\\sinx=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k2\pi\\x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Rightarrow x=\left\{\dfrac{\pi}{2};\dfrac{7\pi}{6};\dfrac{11\pi}{6};\dfrac{5\pi}{2}\right\}\)
\(\Rightarrow\sum x=6\pi\)