\(\cos5x+2\sin x\cos x+2\sin3x\sin2x=0\)
K
Khách
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Những câu hỏi liên quan
28 tháng 6 2021
1.Pt \(\Leftrightarrow cos\left(2x-\dfrac{\pi}{3}\right)=sin\left(x+\dfrac{\pi}{3}\right)\)
\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{3}\right)=cos\left(\dfrac{\pi}{6}-x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{3}=\dfrac{\pi}{6}-x+k2\pi\\2x-\dfrac{\pi}{3}=x-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)\(\left(k\in Z\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k2\pi}{3}\\x=\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)\(\left(k\in Z\right)\)
\(\Rightarrow x=\dfrac{\pi}{6}+\dfrac{k2\pi}{3}\)\(\left(k\in Z\right)\)
2.\(sin^22x+cos^23x=1\)
\(\Leftrightarrow\dfrac{1-cos4x}{2}+\dfrac{1+cos6x}{2}=1\)
\(\Leftrightarrow cos6x=cos4x\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{k\pi}{5}\end{matrix}\right.\)\(\left(k\in Z\right)\)\(\Rightarrow x=\dfrac{k\pi}{5}\)\(\left(k\in Z\right)\) (Gộp nghiệm)
Vậy...
3. \(Pt\Leftrightarrow\left(sinx+sin3x\right)+\left(sin2x+sin4x\right)=0\)
\(\Leftrightarrow2.sin2x.cosx+2.sin3x.cosx=0\)
\(\Leftrightarrow2cosx\left(sin2x+sin3x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sin3x=-sin2x\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\sin3x=sin\left(\pi+2x\right)\end{matrix}\right.\)(\(k\in Z\))
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\pi+k2\pi\\x=\dfrac{k2\pi}{5}\end{matrix}\right.\)(\(k\in Z\))\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{k2\pi}{5}\end{matrix}\right.\) (\(k\in Z\))
Vậy...
4. Pt\(\Leftrightarrow\dfrac{1-cos2x}{2}+\dfrac{1-cos4x}{2}=\dfrac{1-cos6x}{2}\)
\(\Leftrightarrow cos2x+cos4x=1+cos6x\)
\(\Leftrightarrow2cos3x.cosx=2cos^23x\)
\(\Leftrightarrow\left[{}\begin{matrix}cos3x=0\\cosx=cos3x\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k\pi}{3}\\x=-k\pi\\x=\dfrac{k\pi}{2}\end{matrix}\right.\)\(\left(k\in Z\right)\)\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k\pi}{3}\\x=\dfrac{k\pi}{2}\end{matrix}\right.\)\(\left(k\in Z\right)\)
Vậy...
HP
23 tháng 8 2021
2.
\(sin3x+cos2x=1+2sinx.cos2x\)
\(\Leftrightarrow sin3x+cos2x=1+sin3x-sinx\)
\(\Leftrightarrow cos2x+sinx-1=0\)
\(\Leftrightarrow-2sin^2x+sinx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sinx=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
HP
23 tháng 8 2021
1.
\(cos3x-cos4x+cos5x=0\)
\(\Leftrightarrow cos3x+cos5x-cos4x=0\)
\(\Leftrightarrow2cos4x.cosx-cos4x=0\)
\(\Leftrightarrow\left(2cosx-1\right)cos4x=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{1}{2}\\cos4x=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\4x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\end{matrix}\right.\)
NV
Nguyễn Việt Lâm
Giáo viên
12 tháng 8 2021
\(cos5x+cosx-\left(sin6x+sin2x\right)=0\)
\(\Leftrightarrow2cos3x.cos2x-2sin4x.cos2x=0\)
\(\Leftrightarrow2cos2x\left(cos3x-sin4x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\\cos3x=sin4x=cos\left(\dfrac{\pi}{2}-4x\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{\pi}{2}+k\pi\\3x=\dfrac{\pi}{2}-4x+k2\pi\\3x=4x-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
\(\Rightarrow...\)
cos5x+2sinxcosx+2sin3xsin2x=0
⇔cos5x+2sinxcosx+\(\dfrac{1}{2}\)(cosx-cos5x)*2=0
⇔cos5x+2sinxcosx+cosx-cos5x=0
⇔cosx(1+2sinx)=0
⇔cosx=0 hoặc sinx=\(\dfrac{-1}{2}\)
⇔x=\(\dfrac{\Pi}{2}+k\Pi\) hoặc x=\(\dfrac{-1}{6}\Pi+k2\Pi\) hoặc x=\(\dfrac{7}{6}\Pi+k2\Pi\) với k∈Z
Lời giải:
\(\cos 5x+2\sin x\cos x+2\sin 3x\sin 2x=0\)
\(\Leftrightarrow \cos (3x+2x)+2\sin x\cos x+2\sin 3x\sin 2x=0\)
\(\Leftrightarrow \cos 3x\cos 2x-\sin 3x\sin 2x+2\sin x\cos x+2\sin3x\sin 2x=0\)
\(\Leftrightarrow (\cos 3x\cos 2x+\sin 3x\sin 2x)+2\sin x\cos x=0\)
\(\Leftrightarrow \cos (3x-2x)+2\sin x\cos x=0\)
\(\Leftrightarrow \cos x(1+2\sin x)=0\)
\(\Rightarrow \left[\begin{matrix} \cos x=0\\ 1+2\sin x=0\end{matrix}\right.\Rightarrow \left[\begin{matrix} \cos x=0\\ \sin x=\frac{-1}{2}\end{matrix}\right.\)
\(\Rightarrow \left[\begin{matrix} x=k\pi+\frac{\pi}{2}\\ x=\frac{-\pi}{6}+2k\pi\\ x=\frac{7\pi}{6}+2k\pi\end{matrix}\right.\) (k nguyên)