\documentclass[12pt,a4paper]{report} \usepackage[utf8x]{vietnam} \usepackage{ucs} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb,enumerate} \begin{document} \tableofcontents \chapter{HÀM SỐ LƯỢNG GIÁC, PHƯƠNG TRÌNH LƯỢNG GIÁC} \section{HÀM SỐ LƯỢNG GIÁC } \subsection{Tìm miền xác định của hàm số} \begin{enumerate} \item $y=\dfrac{1+\sin x}{\cos x}$ \item $y=\sqrt{\dfrac{1+\sin x}{1-\sin x}}$ \item $y=\cot{(x-\dfrac{\pi}{6})}$ \item $y=\tan{(2x+\dfrac{\pi}{3})}$ \item $y=\sin{\sqrt{\dfrac{1+x}{1-x}}}$ \item $y=\cos{\sqrt{x}}$ \item $y=\cos{\dfrac{1}{x+1}}$ \item $y=\dfrac{2}{3\sin x}$ \end{enumerate} \subsection{ Xét tính chẵn lẻ của hàm số} \begin{enumerate} \item $y=x\cos 2x$ \item $y=x^5\sin x$ \item $y=x-\sin x$ \item $y=\dfrac{1+\cos x}{1-\cos x}$ \item $y=\dfrac{\sin x}{3x}$ \item $y=\dfrac{\cos 3x}{x}$ \end{enumerate} \subsection{ Tìm giá trị lớn nhất nhỏ nhất} \begin{enumerate} \item $y=1-3|\sin 3x|$ \item $y=\cos 3x+\sin(3x+\dfrac{\pi}{3})$ \item $y= \sin^2 x+3\cos 2x$ \item $y=\sqrt{5-\sin^2 x}$ \item $y=3\sqrt{\sin 5x}-2$ \item $y=2-3\cos^2 x$ \end{enumerate} \section{PHƯƠNG TRÌNH LƯỢNG GIÁC CƠ BẢN } \subsection{Giải các phương trình sau:} \begin{enumerate}[\it 1)] \item $\sin 3x=-\dfrac{\sqrt{3}}{2}$ \item $2\sin{(x+\dfrac{\pi}{3})}+\sqrt{3}=0$ \item $\sin (2x-15^0)=\dfrac{\sqrt{2}}{2}$ \item $\sin \Big(\dfrac{x}{2}+10^0\Big)=-\dfrac{1}{2}$ \item $\sin 4x =\dfrac{2}{3}$ \item $\cos (x+3)=\dfrac{1}{3}$ \item $\cos (3x-45^0)=\dfrac{\sqrt{3}}{2}$ \item $\cos{(2x+50^0)}=-\dfrac{\sqrt{2}}{2}$ \item $\cos (x-2)=\dfrac{2}{5}$ \item $\cos \Big(2x+\dfrac{\pi}{3}\Big)=-\dfrac{1}{2}$ \item $(2+\sin x)(3\sin 2x-1)=0$ \item $(1+2\cos x)(3-\cos x)=0$ \item $\tan (2x+45^0)=-1$ \item $\cot \Big(x+\dfrac{\pi}{3}\Big)=\sqrt{3}$ \item $\tan \Big(\dfrac{x}{2}-\dfrac{\pi}{4}\Big)=\tan \dfrac{\pi}{8}$ \item $\cot \Big(\dfrac{x}{3}+20^0\Big)=-\dfrac{\sqrt{3}}{3}$ \item $\sqrt{3}\tan{\Big(x+\dfrac{\pi}{4}\Big)}+1=0$ \end{enumerate} \subsection{Giải các phương trình sau:} \begin{enumerate}[\it 1)] \item $\sin 3x=\sin \Big(x+\dfrac{\pi}{4}\Big)$ \item $\sin \Big(3x-\dfrac{\pi}{4}\Big)=\sin \Big(x+\dfrac{\pi}{6}\Big)$ \item $\cos(2x+1)=\cos(x-2)$ \item $\cos \Big(2x-\dfrac{\pi}{3}\Big)=\cos \Big(\dfrac{\pi}{4}-x\Big)$ \item $\tan 3x=\tan \Big(\dfrac{\pi}{3}-2x\Big)$ \item $\tan \Big(2x+\dfrac{\pi}{5}\Big)=\tan \Big(\dfrac{\pi}{5}-x\Big)$ \item $\cot 3x=\cot \Big(x+\dfrac{\pi}{3}\Big)$ \end{enumerate} \subsection{Giải các phương trình sau:} \begin{enumerate}[\it 1)] \item $\cos 3x-\sin 2x=0$ \item $\tan x\tan 2x=-1$ \item $\sin 3x+\sin 5x=0$ \item $\cot 2x.\cot 3x=1$ \item $\sin 3x=\cos^2 2x-\sin^2 2x$ \item $\sin 2x+\sin{\Big(x-\dfrac{\pi}{4}\Big)}=0$ \item $\sin 5x+2\sin^2{3x}=1$ \item $\cos^2 3x+\sin^2 2x=1$ \item $\sin^2{\Big(x+\dfrac{\pi}{4}\Big)}-\cos^2{\Big(x+\dfrac{\pi}{3}\Big)}=0$ \item $2\cos^2 x=1-\sin 4x$ \item $\sin \Big/(x+\dfrac{2\pi}{3}\Big)=\cos 3x$ \item $\sin \Big(3x-\dfrac{5\pi}{6}\Big)+\cos \Big(3x+\dfrac{\pi}{4}\Big)=0$ \item $\cos \dfrac{x}{2}=-\cos (2x-30^0)$ \item $\cos 2x=\cos x$ \item $\sin{\Big(\dfrac{\pi}{4}+x\Big)}=\sin \Big(2x-\dfrac{\pi}{4}\Big)$ \item $\cot 3x+\cot x=0$ \item $\cot {x}.\tan{\Big(2x-\dfrac{\pi}{6}\Big)}$ \end{enumerate} \subsection{Giải các phương trình sau:} \begin{enumerate}[\it 1)] \item $\sin 2x.\cot x=0$ \item $\tan (x-30^0)\cos (2x-150^0)=0$ \item $(3\tan x+\sqrt{3})(2\sin x-1)=0$ \item $\dfrac{\sin 3x}{\cos 3x-1}=0$ \item $\cos 2x\cot \Big(x-\dfrac{\pi}{4}\Big)=0$ \item $\tan (2x+60^0)\cos (x+75^0)=0$ \item $(\cot x+1)\sin 3x=0$ \end{enumerate} \subsection{Giải các phương trình sau:} \begin{enumerate}[\it 1)] \item $\sin 2x-2\cos x=0$ \item $8\cos 2x\sin 2x\cos 4x=\sqrt{2}$ \item $\tan 2x-2\tan x=0$ \item $2\cos^2 x+\cos 2x=2$ \item $\cos 2x-\sin 2-1=0$ \item $\cos x\cos 2x=1+\sin x\sin 2x$ \item $4\sin x\cos x\cos 2x=-1$ \item $\tan x=3\cot x$ \end{enumerate} \section{PHƯƠNG TRÌNH LƯỢNG GIÁC BẬC NHẤT ĐÓI VỚI $\sin x$ VÀ $\cos x$} \subsection{Giải các phương trình} \begin{enumerate} \item $\sin x+\cos x=\dfrac{\sqrt{6}}{2}$ \item $\sin 2x+\sqrt{3}\cos 2x=1$ \item $\sqrt{3}\cos 3x+\sin 3x=\sqrt{2}$ \item $\sqrt{3}\cos x+\sin x=-2$ \item $\cos 3x-\sin 3x=1$ \item $2\cos x-\sin x=2$ \item $\sin5x+\cos 5x=-1$ \item $3\sin x-4\cos x=1$ \item $5\cos x-12\sin x=1$ \item $\sqrt{2}\cos 3x-\sqrt{6}\cos 3x+\sqrt{6}=0$ \item $5\sin x+6\cos x=7$ \item $\cos 7x-\sqrt{3}\sin 7x=-\sqrt{2}$ \item $\sin x-\cos x=\dfrac{1-\sqrt{3}}{2}$ \item $\sqrt{3}\cos x+\sin x=\sqrt{2}$ \item $2\sin x-\sqrt{3}\cos x=-1$ \item $\sqrt{2}\cos 6x+\sqrt{2}\sin 6x=-1$ \item $3\sin x-5\cos x=\sqrt{17}$ \item $\sqrt{3}\sin 3x-\cos 3x=-1$ \item $3\cos x+7\sin x+5=0$ \item $12\cos x-5\sin x=-13$ \item $27\cos x+5\sin x-15=0$ \end{enumerate} \subsection{Giải các phương trình} \begin{enumerate} \item $\cos x-\sqrt{3}\sin x=2\cos 2x$ \item $\sin 2x-\sqrt{3}\cos 2x=2\sin x$ \item $\sin 8x+\sqrt{3}\cos 7x=\sin 7x+\sqrt{3}\cos 8x$ \item $\cos 8x -\sin 6x=\sqrt{3}(\sin8x+\cos 6x)$ \item $2(\sin^3 x+\cos^3 x)+\sin 2x(\sin x+\cos x)=0$ \item $\sqrt{2}(\cos^4 x-\sin^4 x)=\sin x+\cos x$ \item $(2-\sqrt{3})\cos 5x+\sin 5x=(\sqrt{6}-\sqrt{2})\cos \Big(7x+\dfrac{\pi}{3}\Big)$ \item $(\sqrt{5}-1)\cos 2x+\sqrt{10+2\sqrt{5}}\sin 2x=4\cos \Big(x+\dfrac{\pi}{5}\Big)$ \item $\sqrt{3}\sin 7x-\cos 7x=2\sin \Big(5x-\dfrac{\pi}{6}\Big)$ \item $\cos x-\sqrt{3}\sin x+2=4\sin^2 5x$ \item $\sin 2x+\sqrt{3}\cos 2x=\dfrac{\sqrt{2}}{2}+3\cos \Big(\dfrac{pi}{6}-2x\Big)$ \end{enumerate} \subsection{Tìm $m$ để phương trình có nghiệm} \begin{enumerate} \item $2\sin x+m\cos x=1-m$ \item $3\sin2x-(m+1)\cos 2x=1+3m$ \item $m\sin x+(m+2)\cos x=m-2$ \item $\sqrt{3}\sin 3x-m\cos 3x=2m+1$ \item $(m-1)\cos x+2m\sin x=2$ \item $4m\cos x+(1-2m)\sin x=2-3m$ \item $(m-1)\cos 3x+m\sin 3x+m+1=0$ \item $m\cos 3x+\sin 3x=1+2m$ \item $(m^2-1)\cos x+m\sin x=1+2m$ \item $m\sqrt{m}\cos x+\sqrt{5}\sin x=(m-1)\sqrt{6}$ \item $m\sin x+\cos x=m$ \item $m\cos x+\sqrt{1-m^2}\sin x=2m^2-1$ \end{enumerate} \section{ PHƯƠNG TRÌNH LƯỢNG GIÁC BẬC HAI ĐÓI VỚI MỘT HÀM SỐ LƯỢNG GIÁC} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $2\sin^2 x+5\sin x-3=0$ \item $16\sin^2 x-6\sin x-7=0$ \item $4\cos^2 x-2(\sqrt{3}-1\cos x-\sqrt{3}=0$ \item $10\cos^2+7\cos x+1=0$ \item $12\cos^2 x+\sin x-11=0$ \item $4\sin^2 x-4\cos x-1=0$ \item $9\sin^2 x+9\cos x-5=0$ \item $6\cos^2 x+5\sin x-7=0$ \item $2\sin^2 2x+\sqrt{3}\cos 2x+1=0$ \item $17\sin 2x-\sqrt{17}\cos 4x=0$ \item $\sin^2 2x-\cos^2 x+\dfrac{3}{4}=0$ \item $3\sin^22x+7\cos2x-3=0$ \item $6\cos^2x+5\sin x-7=0$ \item $\cos 2x-5\sin x-3=0$ \item $\cos2x+\cos x+1=0$ \item $6\sin^23x+\cos12x=14$ \item $4\sin^4x+12\cos^2x=7$ \item $2\cos^2x-\sin^2x-4\cos +2=0$ \item $\cos2x+\sin^2x+2\cos x+1=0$ \item $9\sin^2x-5\cos^2x-5\sin x+4=0$ \item $3\cos2x+2(1+\sqrt2+\sin x)\sin x-3-\sqrt2=0$ \item $\cos2x\sin^2x+2\cos x+1=0$ \item $4\sin^22x+8\cos^2x-9=0$ \item $1-5\sin x+2\cos^2x=0$ \item $5-4\sin^2x-8\cos^2\dfrac{x}{2}=-4$ \item $\cos2x+5\sin x+2=0$ \item $4+\cos 2x=3\sin x$ \item $\cos \dfrac{x}{4}-\sqrt{8}\cos \dfrac{x}{8}=0$ \item $\sqrt{17}\sin \dfrac{3x}{2}-\cos 3x=0$ \item $\cos \Big(2x+\dfrac{2\pi}{3}\Big)+4\sin \Big(x+\dfrac{\pi}{3}\Big)=\dfrac{5}{2}$ \item $\cos^2 2x-5\sin^2 x+1=0$ \item $4\cos^2 6x+16\cos^2 3x=13$ \item $(3\sqrt{2}+\sqrt{6})\sin \dfrac{x}{2}+3+\sqrt{3}=2\cos x$ \end{enumerate} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $\tan^2 x-2\tan x+2\sqrt{3}-3=0$ \item $\cot^2 x-(2+\sqrt{2})\cot x+1+\sqrt{2}=0$ \item $\sqrt{3}\tan^2 x+(\sqrt{3}-1)\tan x-1=0$ \item $\tan^2 x-5\tan x+6=0$ \item $\tan^2 x+(4-\sqrt{2})\tan x+3(1-\sqrt{2})=0$ \item $\tan^2 x-(\sqrt{3}-1)\tan x-\sqrt{3}=0$ \item $3\tan x+\sqrt{3}\cot x-3-\sqrt{3}=0$ \item $2\tan x-3\cot x-2=0$ \item $2\tan x+3\cot x=4$ \item $\dfrac{1}{\sin^2 x}+3\cot x+1=0$ \item $\dfrac{1}{\cos^2 x}+\tan x-3=0$ \end{enumerate} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $\cos 4x+1=\cos 2x$ \item $\cos 2x-3\cos x=\cos^2 \dfrac{x}{2}$ \item $\cos^6 x+\sin^6 x=\sin 2x-\dfrac{3}{4}$ \item $4\cos^2 (6x-2)+16\cos^2(3x-1)=13$ \item $(\sin 2x+\sqrt{3}\cos 2x)^2=\cos \Big(2x-\dfrac{\pi}{6}$ \item $\cos (10x+12)+4\sqrt{2}\sin (5x+6)=5$ \item $(1+\sin 2x)(1-\tan x)=1+\tan x$ \item $\tan 3x+\cos 6x=1$ \item $\sin 2x+4\tan x=\dfrac{9\sqrt{3}}{2}$ \item $\cos 2x+2\sin 2x-\tan x+\dfrac{1}{2}=0$ \item $2\cos 8x+\tan 4x=\dfrac{4}{5}$ \item $\dfrac{\tan x+3}{\tan x-3}=\tan x$ \end{enumerate} \section{PHƯƠNG TRÌNH LƯỢNG GIÁC THUẦN NHẤT BẬC HAI ĐỐI VỚI $\sin x$ VÀ $\cos x$} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $\sin^2 x-3\sin x\cos x+2\cos^2 x=0$ \item $\sin^ x+\sqrt{3}\sin 2x-\cos^2 x+1=0$ \item $3\sin^2 x-\sqrt{3}\sin x\cos x+2\cos^2 x-2=0$ \item $\sin^2 x+\sin x\cos x+2\cos^2 x=4$ \item $\sin^2 x-3\sin x\cos x+1+\cos 2x=0$ \item $3\sin^2 x+5\cos^2 x-2\cos 2x-4\sin 2x=0$ \item $4\cos^2 x+3\sin x\cos x-\sin^2 x=3$ \item $2\sin^2 x-\sin x\cos x-\cos^2 x=2$ \item $4\sin^2 x-4\sin x\cos x+3\cos^2 x=1$ \item $\cos^2 x+2\sin x\cos x+5\sin^2 x=2$ \item $3\cos^2 x-2\sin 2x+\sin^2 x=1$ \item $4\cos^2 x-3\sin x\cos x+3\sin^2 x=1$ \item $2\cos^2 x-3\sin 2x+\sin^2 x=1$ \item $6\sin^2 x+\sin x\cos x-\cos^2 x-2=0$ \item $2\sin^2 x-5\sin x\cos x-8\cos^2 x+2=0$ \item $6\sin^2 x-\sin x\cos x-\cos^2 x=3$ \item $(\sqrt{2}+1)\sin^2 x+(\sqrt2-1)\cos^2 x+\sin 2x=\sqrt{2}$ \item $\sqrt3\sin^2 x+(1-\sqrt3)\sin x\cos x-\cos^2 x=\sqrt3-1$ \item $3\sin^2 x+4\sin x\cos x+5\cos^2 x=6$ \end{enumerate} \subsection{Tìm m để phương trình sau có nghiệm} \begin{enumerate}[\it 1)] \item $\sin^2 x+\sin x\cos x+m\cos^2 x+1=0$ \item $m\sin^2 x+\sin 2x+2\cos^2 x=0$ \item $\cos^2 x-\sin x\cos x-2\sin^2 x-m=0$ \item $\sin^2 x+m\sin 2x+2\cos^2 x+1=0$ \item $(m^2+2)\cos^2 x-4m\sin x\cos x+1=0$ \item $m\sin^2 x+2(m-1)\sin x\cos x+2(6-m)\cos^2 x-2=0$ \item $m\sin^2 x-4\sin 2x+(m+6)\cos^2 x=0$ \item $2\cos^2 x-(m+1)\sin 2x+(m^2+2m-1)\sin^2 x=0$ \item $(m+2)\sin^2 x+2(m+2)\sin x\cos x+(5-2m)\cos^2 x=1$ \end{enumerate} \subsection{Tìm giá trị lớn nhất, nhỏ nhất của các hàm số sau:} \begin{enumerate}[\it 1)] \item $y=3\sin^2 x-2\sin 2x-5\cos^2+2$ \item $y=5\sin^2 x+3\sin x\cos x+\cos^2 x$ \item $y=\sqrt3\sin x\cos x+\sin^2 x-2\cos^2 x$ \item $y=2\sin^2 x+\sin 2x+3\cos^2 x+1$ \end{enumerate} \section{PHƯƠNG TRÌNH LƯỢNG GIÁC ĐƯA VỀ DẠNG TÍCH} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $\sin x+\sin 2x+\sin 3x=0$ \item $\sin 3x+\sin 2x+\sin x=1+\cos x+\cos 2x$ \item $\sin^2 x+\sin^2 2x+\sin^2 3x+\sin^2 4x=2$ \item $\cos 4x+\cos 2x+\sqrt2\cos x=0$ \item $\cos 5x+\cos 3x=\sin 6x-\sin 2x$ \item $\cos 3x-\cos 4x+\cos 5x=0$ \item $\sin 7x-\sin 3x=\cos 5x$ \item $cos^2 x-\sin^2 x=\sin 3x+cos 4x$ \item $\cos 2x-\cos x=2\sin^2 \dfrac{3x}{2}$ \item $4\sin 3x+\sin 5x-2\sin x\cos 2x=0$ \item $\sin 7x+\cos 3x+\sin x=0$ \item $\sin 7x\sin 9x=\sin5x\sin 11x$ \item $\sin^2 x+\sin^2 \dfrac{3x}{2}+\sin^2 2x+\sin^2 dfrac{9x}{2}=2$ \item $sin^4 x+\cos^2 x=\cos 4x$ \item $1+cos x+\cos 2x+\cos 3x=0$ \item $\sin7x\cos 3x=\sin 11x+\cos 9x$ \item $sin^2 (2+3x)+\cos^2 \Big(\dfrac{\pi}{4}+2x\Big)=\cos^2 (2-5x)+\sin^2 \Big(\dfrac{\pi}{4}-6x\Big)$ \item $\cos x+\cos 3x+\cos 5x=0$ \item $\sin 3x\sin 9x=\sin5x\sin 7x$ \item $\cos7x\cos3x=\cos14x\cos10x$ \item $\sin11x\cos6x=\sin9x\cos4x$ \item $\sin x\cos2x+\sin2x\cos5x=\sin3x\cos5x$ \item $\sin^22x+\sin^23x+\sin^24x+\sin^29x=2$ \item $\cos^3x\sin3x+\cos3x\sin^3x=\dfrac{3}{4}$ \item $\cos^3x\cos3x+\sin^3x\sin3x=\sin^35x$ \item $\cos3x+\sin7x=2\sin^2 \Big(\dfrac{\pi}{4}+\dfrac{5x}{2}\Big)-2\cos^2 \dfrac{9x}{2}$ \end{enumerate} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $\tan x+\tan 2x=\tan 3x$ \item $1+\tan x=(1-\tan x)(1+\sin 2x)$ \item $\tan x+\cot 2x=2\cot 4x$ \item $\tan x+\tan 2x=\sin 3x\cos x$ \item $\sin x+\cos x=\dfrac{\cos 2x}{1-\sin 2x}$ \item $\dfrac{1}{\sin 2x}+\dfrac{1}{\cos 2x}=\dfrac{2}{\sin 4x}$ \item $\dfrac{\sin^3 x+\cos^3 x}{2\cos x-\sin x}=\cos 2x$ \item $1+\sin x-\cos x-\sin 2x+2\cos 2x=0$ \item $\sin x-\dfrac{1}{\sin x}=\sin^2 x-\dfrac{1}{\sin^2 x}$ \item $\cos x\tan 3x=\sin 5x$ \end{enumerate} \section{CÁC BÀI TOÁN TỔNG HỢP} \subsection{ Giải các phương trình} \begin{enumerate}[\it 1)] \item $5\Big(\sin x+\dfrac{cos 3x+\sin 3x}{1+2\sin 2x}\Big)=\cos2x+3$ \item $\cot x-1=\dfrac{\cos 2x}{1+\tan x}+\sin^2x-\dfrac{1}{2}\sin2x$ ĐS: $x=\dfrac{\pi}{4}+k\pi$ \item $\cos^23x.\cos2x-\cos^2x=0$ ĐS:$x=\dfrac{k\pi}{2}$ \item $\dfrac{2(\cos^6x+\sin^6x)-\sin x\cos x}{\sqrt2-2\sin x}=0$ ĐS: $x=\dfrac{5\pi}{4}+k2\pi$ \item $(1+\sin^2x)\cos x+(1+\cos^2x)\sin x=1+\sin 2x$ ĐS: $x=-\dfrac{\pi}{4}+k\pi; \dfrac{\pi}{2}+k2\pi; k2\pi$ \item $\dfrac{1}{\sin x}+\dfrac{1}{\sin\Big(x+\dfrac{3\pi}{2}\Big)}=4\sin\Big(\dfrac{7\pi}{4}-x\Big)$ ĐS:$x=-\dfrac{\pi}{4}+k\pi; -\dfrac{\pi}{8}+k\pi; \dfrac{5\pi}{8}+k\pi$ \item $\sin3x-\sqrt{3}\cos3x=2\sin2x$ ĐS:$x=\dfrac{\pi}{3}+k2\pi; \dfrac{4\pi}{15}+k\dfrac{2\pi}{5}$ \item $\dfrac{(1-2\sin x)\cos x}{(1+2\sin x)(1-\sin x)}=\sqrt3$ ĐS: $x=-\dfrac{\pi}{18}+k\dfrac{2\pi}{3}$ \item $(1+2\sin x)^2\cos x=1+\sin x+\cos x$ ĐS: $x=-\dfrac{\pi}{2}+k2\pi; \dfrac{\pi}{12}+k\pi; \dfrac{5\pi}{12}+k\pi$ \item $\sin^23x-\cos^24x=\sin^25x-\cos^26x$ ĐS:$x=\dfrac{k\pi}{9}; \dfrac{k\pi}{2}$ \item $\cot x-\tan x+4\sin2x=\dfrac{2}{\sin2x}$ ĐS: $x=\pm\dfrac{\pi}{3}+k2\pi$ \item $5\sin x-2=3(1-\sin x)\tan^2x$ ĐS:$x=\dfrac{\pi}{6}+k2\pi; \dfrac{5\pi}{6}+k2\pi$ \item $1+\sin x+\cos x+\sin2x+\cos2x=0$ ĐS: $x=-\dfrac{\pi}{4}+k\pi; \pm\dfrac{2\pi}{3}+k2\pi$ \item $\cot x+\sin x\Big(1+\tan x\tan\dfrac{x}{2}\Big)=4$ ĐS:$x=\dfrac{\pi}{12}+k\pi; \dfrac{5\pi}{12}+k\pi$ \item $2\sin^22x+\sin7x-1=\sin x$ ĐS: $x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}; \dfrac{\pi}{18}+k\dfrac{2\pi}{3}; \dfrac{5\pi}{18}+k\dfrac{2\pi}{3}$ \item $\sin^3x-\sqrt3\cos^3x=\sin x\cos^2x-\sqrt3\sin^2x\cos x$ ĐS:$x=\dfrac{\pi}{4}+k\dfrac{\pi}{2}; -\dfrac{\pi}{3}+k\pi$ \item $\sin x+\cos x\sin 2x+\sqrt{3}\cos 3x=2(\cos 4x+\sin^2x)$ ĐS: $x=-\dfrac{\pi}{6}+k2\pi; \dfrac{\pi}{42}+k\dfrac{2\pi}{7}$ \item $(1+2\sin^2x)\cos x=1+\sin x+\cos x$ ĐS:$x=-\dfrac{\pi}{2}+k2\pi; \dfrac{\pi}{12}+k\pi; \dfrac{5\pi}{12}+k\pi$ \item $\cos3x-4\cos2x+3\cos x-4=0$ \item $\sin^2\Big(\dfrac{x}{2}-\dfrac{\pi}{4}\Big)\tan^2x-\cos^2\dfrac{x}{2}=0$ ĐS: $x=\pi+k2\pi; -\dfrac{\pi}{4}+k\pi$ \item $(2\cos x-1)(2\sin x+\cos x)=\sin2x-\sin x$ ĐS: $x=\pm\dfrac{\pi}{3}+k2\pi; -\dfrac{\pi}{4}+k\pi$ \item $\sin^4{x}+\cos^4{ x}+\cos{\Big(x-\dfrac{\pi}{4} \Big)}\sin{ \Big(3x-\dfrac{\pi}{4}\Big)}-\dfrac{3}{2}=0$ ĐS: $x=\dfrac{\pi}{4}+k\pi$ \item $cos 3x+\cos2x-\cos x-1=0$ ĐS:$x=k\pi; \pm\dfrac{2\pi}{3}+k2\pi$ \item $\Big(\sin \dfrac{x}{2}+\cos \dfrac{x}{2}\Big)^2+\sqrt{3}\cos x=2$ ĐS: $x=\dfrac{\pi}{2}+k2\pi; -\dfrac{\pi}{6}+k2\pi$ \item $2\sin x(1+\cos 2x)+\sin 2x=1+2\cos x$ ĐS: $x=\pm\dfrac{2\pi}{3}+k2\pi; \dfrac{\pi}{4}+k\pi$ \item $sqrt3\cos5x-2\sin3x\cos2x-\sin x=0$ ĐS:$x=\dfrac{\pi}{18}+k\dfrac{\pi}{3}; -\dfrac{\pi}{6}+k\dfrac{\pi}{2}$ \item $(1+2\sin x)^2\cos x=1+\sin x+\cos x$ ĐS: $x=-\dfrac{\pi}{2}+k2\pi; \dfrac{\pi}{12}+k\pi; \dfrac{5\pi}{12}+k\pi$ \item $\dfrac{2\sin x+\cos x+1}{\sin x-2\cos x+3}=\dfrac{1}{3}$ \item $\tan x+\cos x-\cos^2x=\sin x\Big(1+\tan x\tan\dfrac{x}{2}\Big)$ \item $2\cos2x-8\cos x+7=\dfrac{1}{\cos x}$ \item $(\cos x+1)\cos2x+2\cos x)-2\sin^2x$ \item $4(\sin3x-\cos2x)=5(\sin x-1)$ \item $\sin3x+\sin x-2\cos^2x=0$ \item $2\sqrt2(\sin x+\cos x)\cos x=3+\cos2x$ \item $3\sin3x-\sqrt3\cos9x=1+4\sin^33x$ \item $\cos7x\cos5x-\sqrt3\sin2x=1-\sin7x\sin5x$ \item $\cos^3x+\sin^3x=\sin x-\cos x$ \item $\sin 3x=2\cos^3x$ \item $1+3\sin 2x=2\tan x$ \item $8\cos^3\Big(x+\dfrac{\pi}{3}\Big)=\cos3x$ \item $\cos^3x-\sin^3x=\cos x-\sin x$ \item $4\sin^3x-10\sin^2x\cos x+6\sin x\cos^2x-\cos^3x=0$ \item $8\cos x=\dfrac{\sqrt3}{\sin x}+\dfrac{1}{\cos x}$ \item $\sqrt2\sin^3\Big(x+\dfrac{\pi}{4}\Big)=2\sin x$ \item $4\sin^3x-\sin x-\cos x=0$ \item $\sin3x=\cos x\cos 2x(\tan^2x+\tan2x)$ \item $\dfrac{\cos2x+3\cot2x+\sin4x}{\cot2x-\cos2x}=2$ \item $\dfrac{(1-\cos x)^2+(1+\cos x)^2}{4(1-\sin x)}-\tan^2x\sin x=\dfrac{1+\sin x}{2}+\tan^2x$ \item $3\sin^2x+\dfrac{1}{2}\sin2x+2\cos^2x=\dfrac{3(\sin^4x+\cos^4x-1)}{\sin^6x+\cos^6x-1}$ \item $\dfrac{\sin^42x+\cos^42x}{\tan\Big(\dfrac{\pi}{4}-x\Big)\tan\Big(\dfrac{\pi}{4}+x\Big)}=\cos^42x$ \item $\dfrac{\sin 5x}{5\sin x}=1$ \item $\dfrac{\cos x-2\sin x\cos x}{2\cos^2x-\sin x-1}=\sqrt3$ \item $\tan x-\sin 2x-\cos 2x+2\Big(2\cos x-\dfrac{1}{\cos x}\Big)=0$ \item $1+\cot2x=\dfrac{1-\cos 2x}{\sin^22x}$ \item $6\sin x-2\cos^3x=\dfrac{5\sin 4x\cos x}{2\cos 2x}$ \item $\dfrac{\sin^4x+\cos^4x}{\sin2x}=\dfrac{1}{2}(\tan x+\cot x)$ \item $2(\sin3x-\cos3x)=\dfrac{1}{\sin x}+\dfrac{1}{\cos x}$ \item $2\sqrt2\sin\Big(x+\dfrac{\pi}{4}\Big)=\dfrac{1}{\sin x}+\dfrac{1}{\cos x}$ \item $\dfrac{1}{\tan x+\cot2x}=\dfrac{\sqrt2(\cos x-\sin x)}{\cot x-1}$ \item $\dfrac{\cot^2x-\tan^2x}{\cos2x}=16(1+\cos4x)$ \item $\sin^4x+\cos^4x=\dfrac{7}{8}\cot\Big(x+\dfrac{\pi}{3}\Big)\cot\Big(\dfrac{\pi}{6}-x\Big)$ \item $\tan^4x+1=\dfrac{(2-\sin^22x)\sin3x}{\cos^4x}$ \item $\dfrac{\sin^4x+\cos^4x}{5\sin2x}=\dfrac{1}{2}\cot2x-\dfrac{1}{8\sin2x}$ \item $\sqrt{\dfrac{1}{8\cos^2x}}=\sin x$ \item $\cos2x+\cos x(2\tan x-1)=2$ \item $3-\tan x(\tan x+2\sin x)+6\cos x=0$ \item $3\cos 4x-8\cos^6x+2\cos^2x+3=0$ \item $\dfrac{(2-\sqrt3)\cos x-2\sin^2\Big(\dfrac{x}{2}-\dfrac{\pi}{4}\Big)}{2\cos x-1}=1$ \item $\dfrac{\cos^2x(\cos x-1)}{\sin x+\cos x}=2(1+\sin x)$ \item $\cot x=\tan x+\dfrac{2\cos4x}{\sin2x}$ \item $4(\sin^3x+\cos^3x)=\cos x+3\sin x$ \item $\sqrt{1-\sin x}+\sqrt{1-\cos x}=1$ \item $\sin4x\sin7x=\cos3x\cos6x$ \item $2\sin x\cos2x+\sin2x\cos x=\sin4x\cos x$ \item $\sin x+\sin2x=\sqrt3(\cos x+\cos2x)$ \item $2\sqrt2\cos^3\Big(x-\dfrac{\pi}{4}\Big)-3\cos x-\sin x=0$ \item $\tan\Big(\dfrac{3\pi}{2}-x\Big)+\dfrac{\sin x}{1+\cos x}=2$ \item $\sin2x+\cos2x+3\sin x-\cos x-2=0$ \item $4\sin^2\dfrac{x}{2}-\sqrt3\cos 2x=1+2\cos^2\Big(x-\dfrac{3\pi}{4}$ \item $\sin x\cos2x+\cos^2x(\tan^2x-1)+2\sin^3x=0$ \item $\tan\Big(\dfrac{\pi}{2}+x\Big)-3\tan^3x=\dfrac{\cos2x-1}{\cos^2x}$ \item $\cos3x\cos^3x-\sin3x\sin^3x=\dfrac{2+3\sqrt2}{8}$ \item $2\sin\Big(2x-\dfrac{\pi}{6}\Big)+4\sin x+1=0$ \item $(2\sin^2x-1)\tan^22x+3(2\cos^2x-1)=0$ \item $\cos2x+(1+2\cos x)(\sin x-\cos x)=0$ \item $\sin^3x+\cos^3x+2\sin^2x=1$ \item $4\sin^3x+4\sin^2x+3\sin2x+6\cos x=0$ \item $\sin2x+\sin x-\dfrac{1}{\sin2x}-\dfrac{1}{2\sin x}=2\cot2x$ \item $2\cos^2x+2\sqrt3\sin x\cos x+1=3(\sin x+\sqrt3\cos x)$ \item $\sin\Big(\dfrac{5x}{2}-\dfrac{\pi}{4}\Big)-\cos\Big(\dfrac{x}{2}-\dfrac{\pi}{4}\Big)=\sqrt2\cos\dfrac{3x}{2}$ \item $\dfrac{\sin2x}{\cos x}+\dfrac{\cos2x}{\sin x}=\tan x-\cot x$ \item $2\sqrt2\sin\Big(x-\dfrac{\pi}{12}\Big)\cos x=1$ \item $(1-\tan x)(1+\sin2x)=1+\tan x$ \item $\tan x=\cot x+4\cos^22x$ \item $\sin\Big(2x-\dfrac{\pi}{4}\Big)=\sin\Big(x-\dfrac{\pi}{4}\Big)+\dfrac{\sqrt3}{2}$ \item $2\sin\Big(x+\dfrac{\pi}{3}\Big)-\sin\Big(2x-\dfrac{\pi}{6}\Big)=\dfrac{1}{2}$ \item $3\sin x+\cos2x+\sin2x=4\sin x\cos\dfrac{x}{2}$ \item $4(\sin^4x+\cos^4x)+\cos 4x+\sin2x=0$ \item $2\sqrt2\cos2x+\sin 2x\cos\Big(x+\dfrac{3\pi}{4}\Big)-4\sin\Big(x+\dfrac{\pi}{4}\Big)=0$ \item $\sin^23x-\cos^24x=\sin^25x-\cos^26x$ \item $4\sin^2\Big(\pi-\dfrac{x}{2}\Big)-\sqrt3\sin\Big(\dfrac{\pi}{2}-2x\Big)=1+2\cos^2\Big(x-\dfrac{3\pi}{4}\Big)$ \item $\sin2x+\sin x-\dfrac{1}{2\sin x}-\dfrac{1}{\sin2x}=2\cot 2x$ \item $\dfrac{3\sin2x-2\sin x}{\sin2x\cos x}=2$ \item $\cos2x+5=2(2-\cos x)(\sin x-\cos x)$ \item $9\sin x+6\cos x -3\sin2x+\cos2x=8$ \item $\dfrac{(\sin2x-\sin x+4)\cos x-2}{2\sin x+\sqrt3}=0$ \item $\cos^23x\cos2x-\cos^2x=0$ \item $\dfrac{3\sin2x-\sin 2x}{\sin2x\cos x}=2$ \item $4\cos^4x-\cos2x-\dfrac{1}{2}\cos4x+\cos \dfrac{3x}{4}=\dfrac{7}{2}$ \item $\dfrac{\cos^2{x}(\cos x-1)}{\sin x+\cos x}=2(1+\sin x)$ \item $1+\sin{ \dfrac{x}{2}}\sin x-\cos{\dfrac{x}{2}}\sin^2x=2\cos^2{ \Big(\dfrac{\pi}{4}-\dfrac{x}{2}\Big)}$ \item $4\cos{\dfrac{5x}{2}}\cos{\dfrac{3x}{2}}+2(8\sin x-1)\cos x=5$ \item $(\sin2x+\cos2x)\cos x+2\cos2x-\sin x=0$ \item $\sin2x-\cos2x+3\sin x-\cos x-1=0$ \item $\sin^3{x}(1+\cot x)+\cos^3{x}(1+\tan x)=\sqrt{2\sin2x}$ \item $\sin \Big(3x-\dfrac{\pi}{4}\Big)=\sin2x\sin \Big(x+\dfrac{\pi}{4}\Big)$ \item $\cos^2x+\cos x+\sin^3x=0$ \item $\cos3x-\cos2x+\cos x=\dfrac{1}{2}$ \item $\tan{\Big(x-\dfrac{\pi}{6}\Big)}\tan{\Big(x+\dfrac{\pi}{3}\Big)}\sin3x=\sin x+\sin2x$ \item $2\cos x+\dfrac{1}{3}\cos^2{(x+3\pi)}=\dfrac{8}{3}+\sin{2(x-\pi)}+3\cos{\Big(x+\dfrac{21\pi}{2}\Big)}+\dfrac{1}{3}\sin^2x$ \item $\dfrac{\sqrt{2}\sin{\Big(\dfrac{\pi}{4}-x\Big)}}{\cos x}(1+\sin2x)=1+\tan x$ \item $\dfrac{\sin^3{x}\sin{3x}+\cos^3{x}\cos{3x}}{\tan{\Big(x-\dfrac{\pi}{6}\Big)}\tan{\Big(x+\dfrac{\pi}{3}\Big)}}=-\dfrac{1}{8}$ \item $\tan^2x-\tan^2x\sin^3x+\cos^3x-1=0$ \item $\dfrac{\sin^6x+\cos^6x}{\cos^2x-\sin^2x}=\dfrac{1}{4}\tan2x$ \item $\dfrac{(1+\sin x+\cos2x)\sin {\Big(x+\dfrac{\pi}{4}\Big)}}{1+\tan x}=\dfrac{1}{\sqrt{2}}\cos x$ \end{enumerate} \chapter{TỔ HỢP, XÁC SUẤT} \section{QUY TẮC ĐẾM} \subsection{Quy tắc cộng} \begin{enumerate}[{Bài} \it1.] \item Trong lớp có 18 nam và 12 nữ. Hỏi có nhiêu cách chọn 1 bạn làm lớp trưởng. \item Trên giá có 10 sách Toán, 13 sách Lý, 17 sách Hóa. Hỏi có bao nhiêu cách chọn 1 quyển sách. \item Từ các số 1,2,3,4,5,6,7,8,9 có bao nhiêu cách chọn một số hoặc là số chẵn hoặc là số nguyên tố? \item Lớp có 40 học sinh đăng kí chới ít nhất 1 trong môn bóng đs và cầu lông. Có 30 em dăng kí bóng đá và 25 em đăng kí cầu lông. Hỏi có bao nhiêu em đăng kí cả hai môn thể thao. \end{enumerate} \subsection{Quy tắc nhân } \begin{enumerate} \item \end{enumerate} \section{TỔ HỢP} \section{BIẾN CỐ VÀ XÁC SUẤT} \chapter{DÃY SỐ, CẤP SỐ CỘNG VÀ CẤP SỐ NHÂN} \section{PHƯƠNG PHÁP QUY NẠP TOÁN HỌC} \section{DÃY SỐ} \section{CẤP SỐ CỘNG} \section{CẤP SỐ NHÂN} \chapter{GIỚI HẠN} \section{GIỚI HẠN DÃY SỐ} \subsection{Tính các giới hạn sau} \begin{enumerate} \item $\lim \dfrac{n^2+2n}{3n^2+n+1}$ \item $\lim \dfrac{n^4}{(n+1)(2-n)(n^2+1)}$ \item $\lim \dfrac{n\sqrt[3]{n^3+2}-4}{5n^2+1}$ \item $\lim \dfrac{(2n\sqrt{n}+1)(\sqrt{n}+3)}{(n+1)(n-2)}$ \item $\lim \dfrac{(n+1)(3n^2+2n-1)}{4n^3-5n}$ \item $\lim \dfrac{(2n-1)(3n+2)(4n-3)}{5n^4+7}$ \item $\lim \dfrac{\sqrt[4]{n^4+3n-1}+2n}{4n+3}$ \item $\lim \dfrac{\sqrt[3]{n^3+2n^2}}{2n+1}$ \item $\lim \dfrac{\sqrt{n^2+3}-n-4}{\sqrt{n^2+2}+n}$ \item $\lim \dfrac{(\sqrt{n^2+1}+n)^2}{\sqrt[3]{n^6+n}}$ \item $\lim \dfrac{2n^2+1}{n^3-3n-3}$ \item $\lim \dfrac{2n-1}{(n^2+1)(3n+2)}$ \item $\lim \dfrac{3n^2-5}{(n-3)(2n^2+3)}$ \end{enumerate} \subsection{Tính các giới hạn sau} \begin{enumerate} \item $\lim(\sqrt{n^2+n}-n)$ \item $\lim(\sqrt{n^2+2n}-n+1)$ \item $\lim(\sqrt{n^2+3n}-n-2)$ \item $\lim(\sqrt{4n^2+3n+1}-2n+1)$ \item $\lim(\sqrt{9n^2+5n}-3n+1)$ \item $\lim(\sqrt[3]{n^3+2n^2-n}-n)$ \item $\lim(\sqrt[3]{8n^3+3n^2+1}+1-2n)$ \item $\lim(\sqrt[3]{2n-n^3}+n-1)$ \item $\lim(1+n^2-\sqrt{n^4+3n+1})$ \item $\lim n(\sqrt[3]{n^3+5n}-n)$ \item $\lim n(\sqrt{n^2+1}-\sqrt{n^2-4})$ \end{enumerate} \section{GIỚI HẠN HÀM SỐ} \subsection{Tính các giới hạn sau} \begin{enumerate} \item $\displaystyle\lim_{x \rightarrow 1}\dfrac{x^2-1}{x-1}$ \item $\displaystyle\lim_{x \rightarrow -1}\dfrac{2x^2+3x+1}{-x^2+4x+5}$ \item $\displaystyle\lim_{x \rightarrow 2}\dfrac{x^3+x^2-2x-8}{x^2-3x+2}$ \item $\displaystyle\lim_{x \rightarrow 3}\dfrac{x^3-4x^2+4x-3}{x^2-3x}$ \item $\displaystyle\lim_{x \rightarrow 1}\dfrac{2x^3-3x+1}{x^4-4x+3}$ \item $\displaystyle\lim_{x \rightarrow 1}\dfrac{x^3-2x+1}{x^5-2x+1}$ \item $\displaystyle\lim_{x \rightarrow -3}\dfrac{2x^3+6x^2-x-3}{x^2-9}$ \item $\displaystyle\lim_{x \rightarrow 2}\dfrac{x^2-x-2}{2x^3-3x^2-x-2}$ \item $\displaystyle\lim_{x \rightarrow 1}\dfrac{2x^3-7x^2+4x+1}{x^3+x-2}$ \item $\displaystyle\lim_{x \rightarrow \frac{1}{2}}\dfrac{2x^3-x^2+2x-1}{1-8x^3}$ \item $\displaystyle\lim_{x \rightarrow -3}\dfrac{x^2-9}{-2x^2+5x+33}$ \item $\displaystyle\lim_{x \rightarrow 3}\dfrac{x^2-2x-3}{x^2-3x}$ \end{enumerate} \subsection{Tính các giới hạn sau} \begin{enumerate} \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{x+8}-3}{x^2+2x-3}$ \item $\displaystyle\lim_{x \rightarrow 0} \dfrac{\sqrt{1+2x}-1}{3x}$ \item $\displaystyle\lim_{x \rightarrow 0} \dfrac{4x}{\sqrt{9+x}-3}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{2-\sqrt{x+3}}{x^2-1}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{2x+7}+x-4}{x^2-4x+3}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{x^2+3x}-2}{x-1}$ \item $\displaystyle\lim_{x \rightarrow 2} \dfrac{x-\sqrt{3x-2}}{x^2-4}$ \item $\displaystyle\lim_{x \rightarrow 2} \dfrac{x^2-3x+2}{\sqrt{x+2}-2}$ \item $\displaystyle\lim_{x \rightarrow 0} \dfrac{3-\sqrt{x+9}}{x^2+x}$ \item $\displaystyle\lim_{x \rightarrow 2} \dfrac{x-\sqrt{x+2}}{x^2-3x+2}$ \item $\displaystyle\lim_{x \rightarrow 3} \dfrac{x^2-4x+3}{\sqrt{3x}-3}$ \item $\displaystyle\lim_{x \rightarrow -1} \dfrac{x+1}{\sqrt{6x^2+3}+3x}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{x+8}-3}{1-x^2}$ \item $\displaystyle\lim_{x \rightarrow 7} \dfrac{2-\sqrt{x-3}}{x^2-49}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{3x+1}-2}{1-x^3}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{x+1-\sqrt{x+3}}{x^2-1}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{8x+1}-3x}{x^2-3x+2}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{2x-\sqrt{x+3}}{x^2-6x+5}$ \item $\displaystyle\lim_{x \rightarrow -1} \dfrac{\sqrt{3x+4}-x^2}{x^2+3x+2}$ \item $\displaystyle\lim_{x \rightarrow -2} \dfrac{\sqrt{3x+7}+x+1}{x^2-4}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{\sqrt{7x+2}-x-2}{x^2-3x+2}$ \item $\displaystyle\lim_{x \rightarrow 1} \dfrac{7-5x-\sqrt{x+3}}{x^2-6x+5}$ \item $\displaystyle\lim_{x \rightarrow -1} \dfrac{\sqrt{3x+4}-x^2}{x^2-x-2}$ \item $\displaystyle\lim_{x \rightarrow -2} \dfrac{\sqrt{3x+7}+2x+3}{x^2-4}$ \item $\displaystyle\lim_{x \rightarrow -1} \dfrac{\sqrt{3x+4}+2x+1}{1-x^2}$ \end{enumerate} \section{HÀM SỐ LIÊN TỤC} \chapter{ĐẠO HÀM} \section{ĐỊNH NGHĨA VÀ Ý NGHĨA CỦA ĐẠO HÀM} \section{CÁC QUY TẮC TÍNH ĐẠO HÀM} \section{ĐẠO HÀM CỦA HÀM SỐ LƯỢNG GIÁC} \section{VI PHÂN} \section{ĐẠO HÀM CẤP HAI} \end{document}