`a)3sin^2 x+8sin x cos x+4cos^2 x=0`
`@TH1:cos x=0=>x=\pi/2+k\pi` `(cos x=0=>sin^2 x=1)`
Ptr có dạng: `3sin^2 x=0<=>sin^2 x=0` (Vô lí)
`@TH2:cos x \ne 0<=>x \ne \pi/2+k\pi`
`=>3([sin x]/[cos x])^2+8 [sin x]/[cos x]+4=0`
`<=>3tan^2 x+8tan x+4=0`
`<=>` $\left[\begin{matrix} tan x=\dfrac{-2}{3}\\ tan x=-2\end{matrix}\right.$
`<=>` $\left[\begin{matrix} x=arc tan(\dfrac{-2}{3})+k\pi\\ x=arc tan(-2)+k\pi\end{matrix}\right.$ `(k in ZZ)` (t/m)
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`b)sin^2 x-8sin x cos x+4cos^2 x=0`
`@TH1:cos x=0<=>x=\pi/2+k\pi` `(cos x=0<=>sin^2 x=1)`
Ptr có dạng: `sin^2 x=0` (Vô lí)
`@TH2:cos x \ne 0<=>x \ne \pi/2+k\pi`
`=>([sin x]/[cos x])^2-8[sin x]/[cos x]+4=0`
`<=>tan^2 x-8 tan x+4=0`
`<=>` $\left[\begin{matrix} tan x=4+2\sqrt{3}\\ tan x=4-2\sqrt{3}\end{matrix}\right.$
`<=>` $\left[\begin{matrix} x=arc tan(4+2\sqrt{3})+k\pi\\ x=arc tan(4-2\sqrt{3})+k\pi\end{matrix}\right.$ `(k in ZZ)` (t/m)
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`c)4cos^2 x+3sin x cos x-sin^2 x=3`
`@TH1:cos x=0<=>x=\pi/2+k\pi` `(cos x=0<=>sin^2 x=1)`
Ptr có dạng: `-sin^2 x=3` (Vô lí)
`@TH2:cos x \ne 0<=>x \ne \pi/2+k\pi`
`=>4+3[sin x]/[cos x]-([sin x]/[cos x])^2=3/[cos^2 x]`
`<=>4+3tan x-tan^2 x=3(1+tan^2 x)`
`<=>4tan^2 x-3tan x-1=0`
`<=>` $\left[\begin{matrix} tan x=1\\ tan x=\dfrac{-1}{4}\end{matrix}\right.$
`<=>` $\left[\begin{matrix} x=\dfrac{\pi}{4}+k\pi\\ x=arc tan(\dfrac{-1}{4})+k\pi\end{matrix}\right.$ `(k in ZZ)` (t/m)
a: \(\Leftrightarrow3\cdot\dfrac{1-cos2x}{2}+4\cdot sin2x+4\cdot\dfrac{1+cos2x}{2}=0\)
\(\Leftrightarrow-\dfrac{3}{2}cos2x+\dfrac{3}{2}+4sin2x+2+2cos2x=0\)
\(\Leftrightarrow4sin2x+\dfrac{1}{2}cos2x=-\dfrac{7}{2}\)
\(\Leftrightarrow8sin2x+cos2x=-7\)
\(\Leftrightarrow sin2x\cdot\dfrac{8}{\sqrt{65}}+cos2x\cdot\dfrac{1}{\sqrt{65}}=\dfrac{-7}{\sqrt{65}}\)
Đặt \(cosa=\dfrac{8}{\sqrt{65}}\)
Pt sẽ là \(sin\left(2x+a\right)=\dfrac{-7}{65}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+a=arcsin\left(-\dfrac{7}{65}\right)+k2\Pi\\2x+a=\Pi-arcsin\left(-\dfrac{7}{65}\right)+k2\Pi\end{matrix}\right.\)
=>...
b: \(\Leftrightarrow\left(\dfrac{1}{2}-\dfrac{1}{2}cos2x\right)-4sin2x+4\cdot\dfrac{1+cos2x}{2}=0\)
\(\Leftrightarrow-\dfrac{1}{2}cos2x-4sin2x+\dfrac{1}{2}+2+2cos2x=0\)
\(\Leftrightarrow-4sin2x+\dfrac{3}{2}cos2x=-\dfrac{5}{2}\)
\(\Leftrightarrow8sin2x-3cos2x=5\)
\(\Leftrightarrow sin2x\cdot\dfrac{8}{\sqrt{73}}-cos2x\cdot\dfrac{3}{\sqrt{73}}=\dfrac{5}{\sqrt{73}}\)
\(\Leftrightarrow sin\left(2x-a\right)=\dfrac{5}{\sqrt{73}}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-a=arcsin\left(\dfrac{5}{\sqrt{73}}\right)+k2\Pi\\2x-a=\Pi-arcsin\left(\dfrac{5}{\sqrt{73}}\right)+k2\Pi\end{matrix}\right.\)
=>...
