Câu 23: \(sin2x=-sin\left(x+\Omega\right)\)
=>\(-sin2x=sin\left(x+\Omega\right)\)
=>\(sin\left(-2x\right)=sin\left(x+\Omega\right)\)
=>\(\left[{}\begin{matrix}-2x=x+\Omega+k2\Omega\\-2x=\Omega-x-\Omega+k2\Omega\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}-3x=\Omega+k2\Omega\\-2x=-x+k2\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=-\dfrac{\Omega}{3}-\dfrac{k2\Omega}{3}\\x=-k2\Omega\end{matrix}\right.\)
=>Nghiệm dương nhỏ nhất là \(x=-\dfrac{\Omega}{3}+\dfrac{2\Omega}{3}=\dfrac{\Omega}{3}\)
Câu 24: \(sin2x+cosx=0\)
=>\(2\cdot sinx\cdot cosx+cosx=0\)
=>\(cosx\left(2\cdot sinx+1\right)=0\)
=>\(\left[{}\begin{matrix}cosx=0\\sinx=-\dfrac{1}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\Omega}{2}+k\Omega\\sinx=-\dfrac{1}{2}\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=\dfrac{\Omega}{2}+k\Omega\\x=-\dfrac{\Omega}{6}+k2\Omega\\x=\dfrac{7}{6}\Omega+k2\Omega\end{matrix}\right.\)
\(x\in\left[-2\Omega;2\Omega\right]\)
=>\(\left[{}\begin{matrix}\dfrac{\Omega}{2}+k\Omega\in\left[-2\Omega;2\Omega\right]\\-\dfrac{\Omega}{6}+k2\Omega\in\left[-2\Omega;2\Omega\right]\\\dfrac{7}{6}\Omega+k2\Omega\in\left[-2\Omega;2\Omega\right]\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}k+\dfrac{1}{2}\in\left[-2;2\right]\\2k-\dfrac{1}{6}\in\left[-2;2\right]\\2k+\dfrac{7}{6}\in\left[-2;2\right]\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}k\in\left[-\dfrac{5}{2};\dfrac{3}{2}\right]\\k\in\left[-\dfrac{11}{12};\dfrac{13}{12}\right]\\k\in\left[-\dfrac{31}{12};\dfrac{5}{12}\right]\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}k\in\left\{-2;-1;0;1\right\}\\k\in\left\{0;1\right\}\\k\in\left\{-2;-1;0\right\}\end{matrix}\right.\)
=>Có 4+2+3=9 nghiệm
Câu 25: 2sinx-m-6=0
=>2*sinx=m+6
=>\(sinx=\dfrac{m+6}{2}\)
Để phương trình vô nghiệm thì \(\left[{}\begin{matrix}\dfrac{m+6}{2}< -1\\\dfrac{m+6}{2}>1\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}m+6< -2\\m+6>2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m< -8\\m>-4\end{matrix}\right.\)
23.
\(sin2x=-sin\left(x+\pi\right)\)
\(\Leftrightarrow sin2x=sinx\)
\(\Rightarrow\left[{}\begin{matrix}2x=x+k2\pi\\2x=\pi-x+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=k2\pi\\x=\dfrac{\pi}{3}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
Nghiệm dương nhỏ nhất là \(x=\dfrac{\pi}{3}\)
24.
\(sin2x+cosx=0\)
\(\Leftrightarrow2sinx.cosx+cosx=0\)
\(\Leftrightarrow cosx\left(2sinx+1\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}cosx=0\\sinx=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=-\dfrac{\pi}{3}+k2\pi\\x=\dfrac{4\pi}{3}+k2\pi\end{matrix}\right.\)
\(-2\pi\le\dfrac{\pi}{2}+k\pi\le2\pi\Rightarrow k=\left\{-2;-1;0;1\right\}\) có 4 nghiệm
\(-2\pi\le-\dfrac{\pi}{3}+k2\pi\le2\pi\Rightarrow k=\left\{0;1\right\}\) có 2 nghiệm
\(-2\pi\le\dfrac{4}{3}+k2\pi\le2\pi\Rightarrow k=\left\{-1;0\right\}\) có 2 nghiệm
Tổng cộng có 8 nghiệm
25.
\(2sinx-m-6=0\Rightarrow sinx=\dfrac{m+6}{2}\)
Pt vô nghiệm khi \(\left[{}\begin{matrix}\dfrac{m+6}{2}>1\\\dfrac{m+6}{2}< -1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}m>-4\\m< -8\end{matrix}\right.\)
26.
\(2tanx=\dfrac{m^2}{m-9}\Rightarrow tanx=\dfrac{m^2}{2\left(m-9\right)}\)
Pt có nghiệm khi \(m-9\ne0\Rightarrow m\ne9\)
