\(\Leftrightarrow2cos^2x-1+1-cos^2x+2cosx+1=0\)

\(\Leftrightarrow cos^2x+2cosx+1=0\)

=>cosx=-1

=>x=pi+2kpi

=>(tanx-1)(tanx-4)=0

=>tanx=1 hoặc tanx=4

=>Chọn C

\(tan^2x-5tanx+4=0\Rightarrow\left[{}\begin{matrix}tanx=1\\tanx=4\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\dfrac{1}{tanx}=1\\tanx=4\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}cotx=1\\tanx=4\end{matrix}\right.\)

=>1^2+m^2>=10

=>m^2>=9

=>m>=3 hoặc m<=-3

ĐKXĐ: 3x<>kpi

=>x<>kpi/3

\(PT\Rightarrow sinx\cdot\dfrac{\sqrt{3}}{2}-cosx\cdot\dfrac{1}{2}=-\dfrac{1}{2}\)

=>sin(x-pi/6)=-1/2

=>x-pi/6=-pi/6+k2pi hoặc x-pi/6=7/6pi+k2pi

=>x=k2pi(loại) hoặc x=4/3pi+k2pi(loại)

=>PT không có nghiệm

\(\Leftrightarrow\sqrt{3}sin2x=-sin2x\)

\(\Leftrightarrow2sinx.cosx+\sqrt{3}sinx=0\)

\(\Rightarrow\left[{}\begin{matrix}sinx=0\\cosx=-\dfrac{\sqrt{3}}{2}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=k\pi\\x=\pm\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{-\pi;-\dfrac{7\pi}{6};-\dfrac{5\pi}{6}\right\}\)

\(\Leftrightarrow cos\left(2x+\dfrac{pi}{6}\right)+cos\left(\dfrac{pi}{6}+2x\right)=\sqrt{2}\)

=>cos(2x+pi/6)=căn 2/2

=>2x+pi/6=pi/4+k2pi hoặc 2x+pi/6=-pi/4+k2pi

=>2x=1/2pi+k2pi hoặc 2x=-5/12pi+2kpi

=>x=1/4pi+kpi hoặc x=-5/24pi+kpi

Đặt \(2x+\dfrac{\pi}{6}=t\Rightarrow\dfrac{\pi}{3}-2x=\dfrac{\pi}{2}-t\)

\(\Rightarrow cost+sin\left(\dfrac{\pi}{2}-t\right)=\sqrt{2}\)

\(\Leftrightarrow2cost=\sqrt{2}\)

\(\Rightarrow cost=\dfrac{\sqrt{2}}{2}\)

\(\Rightarrow cos\left(2x+\dfrac{\pi}{6}\right)=\dfrac{\sqrt{2}}{2}\)

\(\Rightarrow\left[{}\begin{matrix}2x+\dfrac{\pi}{6}=\dfrac{\pi}{4}+k2\pi\\2x+\dfrac{\pi}{6}=-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{24}+k\pi\\x=-\dfrac{5\pi}{24}+k\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{\dfrac{\pi}{24};\dfrac{19\pi}{24}\right\}\)

\(cos2x=cos\left(\dfrac{\pi}{2}-x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{\pi}{2}-x+k2\pi\\2x=x-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+\dfrac{k2\pi}{3}\\x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{\dfrac{\pi}{6};\dfrac{5\pi}{6};\dfrac{3\pi}{2}\right\}\)

=>sinx=sin(pi/2-2x)

=>x=pi/2-2x+k2pi hoặc x=pi/2+2x+k2pi

=>x=pi/6+k2pi/3 hoặc x=-pi/2-k2pi

=>\(x\in\left\{\dfrac{pi}{6};\dfrac{5}{6}pi;\dfrac{3}{2}pi\right\}\)

ĐKXĐ: \(x\ne\dfrac{k\pi}{2}\)

Chia 2 vế cho \(cos^2x\) ta được:

\(tan^3x+cotx+2tanx=\dfrac{4\sqrt{3}}{3}\left(1+tan^2x\right)\)

\(\Leftrightarrow\dfrac{\left(tan^2x+1\right)^2}{tanx}=\dfrac{4\sqrt{3}}{3}\left(1+tan^2x\right)\)

\(\Rightarrow tanx=\dfrac{\sqrt{3}}{4}\)

\(\Rightarrow x=arctan\left(\dfrac{\sqrt{3}}{4}\right)+k\pi\)

Tổng nghiệm âm lớn nhất và dương nhỏ nhất là: \(2arctan\left(\dfrac{\sqrt{3}}{4}\right)-\pi\)

Bạn kiểm tra lại đề bài

\(\dfrac{\sqrt{3}}{2}cos2x+\dfrac{1}{2}sin2x=1\)

\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{6}\right)=1\)

\(\Leftrightarrow2x-\dfrac{\pi}{6}=k2\pi\)

\(\Leftrightarrow x=\dfrac{\pi}{12}+k\pi\)

Hiệu giữanghiệm dương nhỏ nhất và âm lớn nhất là:

\(\dfrac{\pi}{12}-\left(\dfrac{\pi}{12}-\pi\right)=\pi\)

\(\Leftrightarrow sin\left(2x+\dfrac{pi}{3}\right)=1\)

=>2x+pi/3=pi/2+k2pi

=>x=pi/12+kpi

Khi k=0 thì x=pi/12

Khi k=-1 thì x=-11/12pi

=->Hiệu là pi/12+11/12pi=pi

\(\Leftrightarrow sin\left(x-\dfrac{pi}{3}\right)=\dfrac{\sqrt{2}}{2}\)

=>x-pi/3=pi/4+k2pi hoặc x-pi/3=3/4pi+k2pi

=>x=7/12pi+k2pi hoặc x=13/12pi+k2pi

Khi k=-1 thì x=7/12pi-2pi=-17/12pi

=>Nghiệm âm lớn nhất là -17/12pi

\(\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{3}=\dfrac{\pi}{4}+k2\pi\\x-\dfrac{\pi}{3}=\dfrac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{7\pi}{12}+k2\pi\\x=\dfrac{13\pi}{12}+k2\pi\end{matrix}\right.\)

Nghiệm âm lớn nhất là \(\dfrac{13\pi}{12}-2\pi=-\dfrac{11\pi}{12}\)