dùng công thức hạ bậc để giải các phương trình sau :
a) \(\sin^24x+\sin^23x=\sin^22x+\sin^2x\)
b) \(\cos^2x+\cos^22x+\cos^23x+\cos^24x=2\)
Những câu hỏi liên quan
\(\sin^2x+sin^23x=\cos^22x+\cos^24x\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{2}cos2x+\frac{1}{2}-\frac{1}{2}cos6x=\frac{1}{2}+\frac{1}{2}cos4x+\frac{1}{2}+\frac{1}{2}cos8x\)
\(\Leftrightarrow cos8x+cos2x+cos6x+cos4x=0\)
\(\Leftrightarrow2cos5x.cos3x+2cos5x.cosx=0\)
\(\Leftrightarrow cos5x\left(cos3x+cosx\right)=0\)
\(\Leftrightarrow2cos5x.cos2x.cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos5x=0\\cos2x=0\\cosx=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{10}+\frac{k\pi}{5}\\x=\frac{\pi}{4}+\frac{k\pi}{2}\\x=\frac{\pi}{2}+k\pi\end{matrix}\right.\)
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Giải các PT sau
1. \(\cos^2\left(x-30^{\cdot}\right)-\sin^2\left(x-30^{\cdot}\right)=\sin\left(x+60^{\cdot}\right)\)
2. \(\sin^22x+\cos^23x=1\)
3. \(\sin x+\sin2x+\sin3x+\sin4x=0\)
4. \(\sin^2x+\sin^22x=\sin^23x\)
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)
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3. \(Pt\Leftrightarrow\left(sinx+sin3x\right)+\left(sin2x+sin4x\right)=0\)
\(\Leftrightarrow2.sin2x.cosx+2.sin3x.cosx=0\)
\(\Leftrightarrow2cosx\left(sin2x+sin3x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sin3x=-sin2x\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\sin3x=sin\left(\pi+2x\right)\end{matrix}\right.\)(\(k\in Z\))
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\pi+k2\pi\\x=\dfrac{k2\pi}{5}\end{matrix}\right.\)(\(k\in Z\))\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{k2\pi}{5}\end{matrix}\right.\) (\(k\in Z\))
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4. Pt\(\Leftrightarrow\dfrac{1-cos2x}{2}+\dfrac{1-cos4x}{2}=\dfrac{1-cos6x}{2}\)
\(\Leftrightarrow cos2x+cos4x=1+cos6x\)
\(\Leftrightarrow2cos3x.cosx=2cos^23x\)
\(\Leftrightarrow\left[{}\begin{matrix}cos3x=0\\cosx=cos3x\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k\pi}{3}\\x=-k\pi\\x=\dfrac{k\pi}{2}\end{matrix}\right.\)\(\left(k\in Z\right)\)\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k\pi}{3}\\x=\dfrac{k\pi}{2}\end{matrix}\right.\)\(\left(k\in Z\right)\)
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Giải phương trình: \(\sin^23x-\cos^24x=\sin^25x-\cos^26x\)
\(sin^23x-cos^24x=sin^25x-cos^26x\)
\(\Leftrightarrow2sin^23x-2cos^24x=2sin^25x-2cos^26x\)
\(\Leftrightarrow2sin^23x-1+1-2cos^24x=2sin^25x-1+1-2cos^26x\)
\(\Leftrightarrow-cos6x-cos8x=-cos10x-cos12x\)
\(\Leftrightarrow cos6x-cos12x+cos8x-cos10x=0\)
\(\Leftrightarrow sin9x.sin6x+sin9x.sin4x=0\)
\(\Leftrightarrow sin9x.\left(sin6x+sin4x\right)=0\)
\(\Leftrightarrow2sin9x.sin5x.cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin9x=0\\sin5x=0\\cosx=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{9}\\x=\dfrac{k\pi}{5}\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
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Giải các phương trình sau:
\(5\sin^22x-6\sin4x-2\cos^2x=0\)
\(2\sin^23x-10\sin6x-\cos^23x=-2\)
\(\sin^2x\left(\tan x+1\right)=3\sin x\left(\cos x-\sin x\right)+3\)
\(6\sin x-2\cos^3x=\frac{5\sin4x.\cos x}{2\cos2x}\)
Giải phương trình:
a) \(Sin^22x+Cos^23x=0\)
b) \(Sin\left(x+\frac{\pi}{3}\right)Cos\left(x-\frac{\pi}{6}\right)=1\)
c) \(Cos^2x+Cos^22x+Cos^23x=1\)
giải pt
a, \(\sin^2x+\sin^22x+\sin^23x=\dfrac{3}{2}\)
b. \(\cos^2x+\sin^22x+\cos^23x=1\)
c,\(\sin5x+2\cos^2x=1\)
d,\(1+\tan x=2\sqrt{2}\sin\left(x+\dfrac{\pi}{4}\right)\)
e,\(\sin3x+\cos3x-\sin x+\cos x=\sqrt{2}\cos2x\)
Giải các phương trình :
a) cos^2x+cos^22x-cos^23x-cos^24x0
b) cos4xcosleft(pi+2xright)-sin2xcosleft(dfrac{pi}{2}-4xright)dfrac{sqrt{2}}{2}sin4x
c) tanleft(120^0+3xright)-tanleft(140^0-xright)2sinleft(80^0+2xright)
d) tan^2dfrac{x}{2}+sin^2dfrac{x}{2}tandfrac{x}{2}+cos^2dfrac{x}{2}+cot^2dfrac{x}{2}+sin x4
e) dfrac{sin2t+2cos^2t-1}{cot t-cot3t+sin3t-sin t}cos t
Đọc tiếp
Giải các phương trình :
a) \(\cos^2x+\cos^22x-\cos^23x-\cos^24x=0\)
b) \(\cos4x\cos\left(\pi+2x\right)-\sin2x\cos\left(\dfrac{\pi}{2}-4x\right)=\dfrac{\sqrt{2}}{2}\sin4x\)
c) \(\tan\left(120^0+3x\right)-\tan\left(140^0-x\right)=2\sin\left(80^0+2x\right)\)
d) \(\tan^2\dfrac{x}{2}+\sin^2\dfrac{x}{2}\tan\dfrac{x}{2}+\cos^2\dfrac{x}{2}+\cot^2\dfrac{x}{2}+\sin x=4\)
e) \(\dfrac{\sin2t+2\cos^2t-1}{\cot t-\cot3t+\sin3t-\sin t}=\cos t\)
Giải phương trình lượng giác sau:
\(sin\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)=cos\left(\dfrac{\pi}{2}-x\right)\)
\(sin^22x=sin^23x\)
a: \(\Leftrightarrow sin\left(\dfrac{x}{3}-\dfrac{pi}{4}\right)=sinx\)
=>x/3-pi/4=x+k2pi hoặc x/3-pi/4=pi-x+k2pi
=>2/3x=-pi/4+k2pi hoặc 4/3x=5/4pi+k2pi
=>x=-3/8pi+k3pi hoặc x=15/16pi+k*3/2pi
b: =>(sin3x-sin2x)(sin3x+sin2x)=0
=>sin3x-sin2x=0 hoặc sin 3x+sin 2x=0
=>sin 3x=sin 2x hoặc sin 3x=sin(-2x)
=>3x=2x+k2pi hoặc 3x=pi-2x+k2pi hoặc 3x=-2x+k2pi hoặc 3x=pi+2x+k2pi
=>x=k2pi hoặc x=pi/5+k2pi/5 hoặc x=k2pi/5 hoặc x=pi+k2pi
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\(cos^2x+cos^22x+cos^23x+cos^24x=2\)
