Giải các phương trình sau :
a) \(3sin^2x-4sinxcosx+5cos^2x=2\)
b) \(25sin^2x+15sin2x+9cos^2=25\)
c) sinx + cosx =1
d) 3cos2x - 4sin2x =1
f) \(4sin^2x-6cos^2x=0\)
g) \(5sin2x-6cos^2x=13\)
h) \(sinx=\sqrt{3}cosx\)
i) \(sin^4x+cos^4\left(x+\frac{\pi}{4}\right)=\frac{1}{4}\)
j)\(tanx+2cotx-3=0\)
k) \(tan^25x=\frac{1}{3}\)
m) \(sin^4x-cos^4x=cosx-2\)
m)
$\sin 4x-\cos ^4x=\cos x-2$
$\Leftrightarrow (\sin ^2x+\cos ^2x)(\sin ^2x-\cos ^2x)=\cos x-2$
$\Leftrightarrow \sin ^2x-\cos ^2x=\cos x-2$
$\Leftrightarrow 1-2\cos ^2x=\cos x-2$
$\Leftrightarrow 2\cos ^2x+\cos x-3=0$
$\Leftrightarrow (2\cos x+3)(\cos x-1)=0$
Nếu $2\cos x+3=0\Rightarrow \cos x=\frac{-3}{2}< -1$ (loại)
Nếu $\cos x-1=0\Rightarrow \cos x=1\Rightarrow x=2k\pi$ với $k$ nguyên
k) ĐK:.......
$\tan ^25x=\frac{1}{3}\Rightarrow \tan 5x=\pm \sqrt{\frac{1}{3}}$
$\Rightarrow 5x=k\pi +\tan ^{-1}\frac{\pm 1}{\sqrt{3}}$
$\Rightarrow x=frac{k}{5}\pi +\tan ^{-1}\frac{\pm 1}{\sqrt{3}}$ với $k$ nguyên.
Số đẹp hơn thì có thể giải như sau:
$PT \Leftrightarrow \frac{\sin ^25x}{\cos ^25x}=\frac{1}{3}$
$\Rightarrow 3\sin ^25x=\cos ^25x$
$\Rightarrow 4\\sin ^25x=1\Rightarrow \sin 5x=\pm \frac{1}{2}$
$\Rightarrow x=\frac{k\pi}{5}\pm \frac{\pi}{30}$ với $k$ nguyên.