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\(sin^2\alpha=1-sin^2\alpha=1-\left(\dfrac{-4}{5}\right)^2=\dfrac{9}{25}\)
vì π<α<\(\dfrac{3\Pi}{2}\)⇒cos α =\(\dfrac{-3}{5}\)
cos2a =1- sin2a =1-\(\left(\dfrac{-4}{5}\right)^2\)=\(\dfrac{3}{5}\)
Vì π<a<\(\dfrac{3\pi}{2}\)
=>cos a =\(\dfrac{-3}{5}\)
\(\cos^2x=\sqrt{1-\dfrac{9}{25}}=\dfrac{16}{25}\)
mà \(\cos x< 0\)
nên \(\cos x=-\dfrac{4}{5}\)
=>\(\tan x=-\dfrac{3}{4};\cot x=-\dfrac{4}{3}\)
Lời giải:
$-\frac{4}{5}=\cos 2x=2\cos ^2x-1$
$\Leftrightarrow \cos ^2x=\frac{1}{10}$
Vì $x\in (\frac{\pi}{4}; \frac{\pi}{2})$ nên $\cos x>0$
$\Rightarrow \cos x=\sqrt{\frac{1}{10}}$
$\sin^2x=1-\cos ^2x=\frac{9}{10}$
Vì $x\in (\frac{\pi}{4}; \frac{\pi}{2})$ nên $\sin x>0$
$\Rightarrow \sin x=\frac{3}{\sqrt{10}}$
$\sin (x+\frac{\pi}{3})=\sin x\cos \frac{\pi}{3}+\cos x\sin \frac{\pi}{3}$
$=\sqrt{\frac{9}{10}}.\frac{1}{2}+\sqrt{\frac{1}{10}}.\frac{\sqrt{3}}{2}=\frac{\sqrt{30}+3\sqrt{10}}{20}$
\(\left\{{}\begin{matrix}x\in\left(0;\dfrac{\pi}{2}\right)\\sinx=\dfrac{\sqrt{3}}{2}\end{matrix}\right.\) \(\Rightarrow x=\dfrac{\pi}{3}\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{\pi}{6}\Rightarrow cos\dfrac{x}{2}=cos\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{2}\)
\(sin^6\left(\pi+x\right)=sin^6x,cos^6\left(x-\pi\right)=cos^6\pi\\ sin^4\left(x+2\pi\right)=sin^4x,sin^4\left(x-\dfrac{3\pi}{2}\right)=cos^4x,cos^2\left(x-\dfrac{\pi}{2}\right)=sin^2x.\)
Khi đó \(A=sin^6x+cos^6x-2sin^4x-cos^4x+sin^2x\\ =\left(sin^2x+cos^2x\right)^2-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)-\left(sin^4x+cos^4x\right)-sin^4x+sin^2x\\ =1-3sin^2x.cos^2x-\left[1-2sin^2x.cos^2x\right]-sin^2x.\left(sin^2x-1\right)\\ =1-3sin^2x.cos^2x-1+2sin^2x.cos^2x+sin^2x.cos^2x\\ =0\)
1+cot^2x=1/sin^2x
=>1/sin^2x=3/2
=>sin^2x=2/3
mà sin x<0
nên sin x=căn 2/3