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ĐKXĐ: \(cosx\ne0\Rightarrow x\ne\dfrac{\pi}{2}+k\pi\)
\(\dfrac{1}{2}cos4x+\dfrac{4sinx}{cosx}.cos^2x=m\)
\(\Rightarrow\dfrac{1}{2}cos4x+2sin2x=m\)
\(\Rightarrow\dfrac{1}{2}\left(1-2sin^22x\right)+2sin2x=m\)
\(\Rightarrow-sin^22x+2sin2x+\dfrac{1}{2}=m\)
Đặt \(sin2x=t\in\left[-1;1\right]\Rightarrow-t^2+2t+\dfrac{1}{2}=m\)
Xét hàm \(f\left(t\right)=-t^2+2t+\dfrac{1}{2}\) trên \(\left[-1;1\right]\)
\(-\dfrac{b}{2a}=1\) ; \(f\left(-1\right)=-\dfrac{5}{2}\) ; \(f\left(1\right)=\dfrac{3}{2}\) \(\Rightarrow-\dfrac{5}{2}\le f\left(t\right)\le\dfrac{3}{2}\)
\(\Rightarrow\) Phương trình đã cho vô nghiệm khi \(\left[{}\begin{matrix}m< -\dfrac{5}{2}\\m>\dfrac{3}{2}\end{matrix}\right.\)
tan(x+pi/6)=-cot(2x-pi/3)
<=>tan(x+pi/6)=tan(pi/2+2x-pi/3)
<=>tan(x+pi/6)=tan(pi/6+2x).........
Bạn tự giải tiếp nha bạn
1.
\(\cos2x+\sin\left(x+\frac{pi}{4}\right)=0\)
\(\Leftrightarrow\sin\left(x+\frac{pi}{4}\right)=-\cos2x\)
\(\Leftrightarrow\sin\left(x+\frac{pi}{4}\right)=\sin\left(2x-\frac{pi}{2}\right)\)
\(\Leftrightarrow\orbr{\begin{cases}x+\frac{pi}{4}=2x-\frac{pi}{2}+k2pi\\x+\frac{pi}{4}=pi-2x+\frac{pi}{2}+k2pi\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}-x=-\frac{3}{4}pi+k2pi\\3x=+\frac{5}{4}pi+k2pi\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{3}{4}pi+k2pi\\x=\frac{5}{12}pi+k\frac{2}{3}pi\end{cases}}\)
2.
\(\sin\left(3x-\frac{5pi}{6}\right)+\cos\left(3x+\frac{3pi}{6}\right)=0\)
\(\Leftrightarrow\sin\left(3x-\frac{5pi}{6}\right)=-\cos\left(3x+\frac{3pi}{6}\right)\)
\(\Leftrightarrow\sin\left(3x-\frac{5pi}{6}\right)=\sin\left(3x+\frac{3pi}{6}-\frac{pi}{2}\right)\)
\(\Leftrightarrow\orbr{\begin{cases}3x-\frac{5pi}{6}=3x+\frac{3pi}{6}-\frac{pi}{2}+k2pi\\3x-\frac{5pi}{6}=pi-3x-\frac{3pi}{6}+\frac{pi}{2}+k2pi\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}0x=\frac{5pi}{6}+k2pi\left(VN\right)\\6x=\frac{11pi}{6}+k2pi\end{cases}}\)
\(\Leftrightarrow x=\frac{11pi}{36}+k\frac{1}{3}pi\)
ĐKXĐ: \(\left\{{}\begin{matrix}x\ne\dfrac{\pi}{2}+k\pi\\x\ne-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{\left(1+2cos^2x-1+2sinx.cosx\right)cosx+cos^2x-sin^2x}{1+\dfrac{sinx}{cosx}}=cosx\)
\(\Leftrightarrow\dfrac{2cos^2x\left(sinx+cosx\right)+\left(sinx+cosx\right)\left(cosx-sinx\right)}{\dfrac{sinx+cosx}{cosx}}=cosx\)
\(\Leftrightarrow\dfrac{cosx\left(sinx+cosx\right)\left(2cos^2x+cosx-sinx\right)}{sinx+cosx}=cosx\)
\(\Rightarrow2cos^2x+cosx-sinx=1\)
\(\Rightarrow cosx-sinx-cos2x=0\)
\(\Rightarrow cosx-sinx-\left(cos^2x-sin^2x\right)=0\)
\(\Rightarrow cosx-sinx-\left(cosx-sinx\right)\left(cosx+sinx\right)=0\)
\(\Rightarrow\left(cosx-sinx\right)\left(1-sinx-cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=cosx\\sin\left(x+\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=k2\pi\\x=\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\) \(\Rightarrow x=\dfrac{\pi}{4}\)
Có 1 nghiệm trên khoảng đã cho
3.
ĐKXĐ: ...
\(\Leftrightarrow tan^22x+\left(\frac{1}{cos^22x}+1\right)=8\)
\(\Leftrightarrow tan^22x+tan^22x=8\)
\(\Leftrightarrow tan^22x=4\)
\(\Rightarrow\left[{}\begin{matrix}tan2x=2\\tan2x=-2\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}2x=arctan\left(2\right)+k180^0\\2x=-arctan\left(2\right)+k180^0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\frac{1}{2}arctan\left(2\right)+k90^0\\x=-\frac{1}{2}arctan\left(2\right)+k90^0\end{matrix}\right.\)
Nghiệm trên nhận các giá trị \(k=\left\{0;1;2;3\right\}\) ; nghiệm dưới nhận các giá trị \(k=\left\{1;2;3;4\right\}\)
1. ĐKXĐ: ...
\(\Leftrightarrow tan\left(x+\frac{\pi}{3}\right)=\frac{1}{tan\left(2x-\frac{\pi}{4}\right)}\)
\(\Leftrightarrow tan\left(x+\frac{\pi}{3}\right)=cot\left(2x-\frac{\pi}{4}\right)\)
\(\Leftrightarrow tan\left(x+\frac{\pi}{3}\right)=tan\left(\frac{3\pi}{4}-2x\right)\)
\(\Leftrightarrow x+\frac{\pi}{3}=\frac{3\pi}{4}-2x+k\pi\)
\(\Rightarrow x=\frac{5\pi}{36}+\frac{k\pi}{3}\)
2.
ĐKXĐ: ...
\(\Leftrightarrow tan\left(x+1\right)=\frac{1}{cot\left(2x+3\right)}\)
\(\Leftrightarrow tan\left(x+1\right)=tan\left(2x+3\right)\)
\(\Leftrightarrow2x+3=x+1+k\pi\)
\(\Rightarrow x=-2+k\pi\)
a.
\(y'=\dfrac{3}{cos^2\left(3x-\dfrac{\pi}{4}\right)}-\dfrac{2}{sin^2\left(2x-\dfrac{\pi}{3}\right)}-sin\left(x+\dfrac{\pi}{6}\right)\)
b.
\(y'=\dfrac{\dfrac{\left(2x+1\right)cosx}{2\sqrt{sinx+2}}-2\sqrt{sinx+2}}{\left(2x+1\right)^2}=\dfrac{\left(2x+1\right)cosx-4\left(sinx+2\right)}{\left(2x+1\right)^2}\)
c.
\(y'=-3sin\left(3x+\dfrac{\pi}{3}\right)-2cos\left(2x+\dfrac{\pi}{6}\right)-\dfrac{1}{sin^2\left(x+\dfrac{\pi}{4}\right)}\)
a1.
$\cot (2x+\frac{\pi}{3})=-\sqrt{3}=\cot \frac{-\pi}{6}$
$\Rightarrow 2x+\frac{\pi}{3}=\frac{-\pi}{6}+k\pi$ với $k$ nguyên
$\Leftrightarrow x=\frac{-\pi}{4}+\frac{k}{2}\pi$ với $k$ nguyên
a2. ĐKXĐ:...............
$\cot (3x-10^0)=\frac{1}{\cot 2x}=\tan 2x$
$\Leftrightarrow \cot (3x-\frac{\pi}{18})=\cot (\frac{\pi}{2}-2x)$
$\Rightarrow 3x-\frac{\pi}{18}=\frac{\pi}{2}-2x+k\pi$ với $k$ nguyên
$\Leftrightarrow x=\frac{\pi}{9}+\frac{k}{5}\pi$ với $k$ nguyên.
a3. ĐKXĐ:........
$\cot (\frac{\pi}{4}-2x)-\tan x=0$
$\Leftrightarrow \cot (\frac{\pi}{4}-2x)=\tan x=\cot (\frac{\pi}{2}-x)$
$\Rightarrow \frac{\pi}{4}-2x=\frac{\pi}{2}-x+k\pi$ với $k$ nguyên
$\Leftrightarrow x=-\frac{\pi}{4}+k\pi$ với $k$ nguyên.
a4. ĐKXĐ:.....
$\cot (\frac{\pi}{6}+3x)+\tan (x-\frac{\pi}{18})=0$
$\Leftrightarrow \cot (\frac{\pi}{6}+3x)=-\tan (x-\frac{\pi}{18})=\tan (\frac{\pi}{18}-x)$
$=\cot (x+\frac{4\pi}{9})$
$\Rightarrow \frac{\pi}{6}+3x=x+\frac{4\pi}{9}+k\pi$ với $k$ nguyên
$\Rightarrow x=\frac{5}{36}\pi + \frac{k}{2}\pi$ với $k$ nguyên.
Đề bài tào lao thật sự
Vừa độ vừa radian trong 1 phương trình là không chính xác. Đã độ thì độ hết, đã radian thì radian hết