Giải các pt sau
1, Sin\(^4\)x + sin\(^4\)(x+\(\frac{\pi}{4}\)) = \(\frac{3}{4}\)
2, Cos\(\frac{x}{2}\) + 2 cos\(^2\)(2x-\(\frac{\pi}{8}\)) =1
3, Sin3x+1=2sin\(^2\)( \(\frac{x}{2}\)+ \(\frac{\pi}{6}\))
4, Sin\(^4\)2x + cos\(^4\)2x =\(\frac{5}{8}\)
5, Cos (3x-\(\frac{\pi}{6}\)) -cos ( 3x + \(\frac{\pi}{6}\)) = \(\frac{\sqrt{2}}{2}\)
32434123421314253646547634764537547352745467524673254-2434217364726465265326546324564527465R632+5483675763547326765476347625374573256732547653274523654763254732654:41625462514651352412436521544E6532655E51263425164652143426543654253421534252344352345
\(1.\sin^4x+\sin^4\left(x+\frac{\pi}{\text{4}}\right)=\frac{3}{4}\)
\(\Leftrightarrow\frac{\left(1-\cos2x\right)^2}{4}+\frac{\left(1-\cos\left(2x+\frac{\pi}{2}\right)\right)^2}{4}=\frac{3}{4}\)
\(\Leftrightarrow\left(1-\cos2x\right)^2+\left(1+\sin2x\right)^2=3\)
\(\Leftrightarrow1-2\cos2x+\cos^22x+1+2\sin2x+\sin^22x=3\)
\(\Leftrightarrow3+2\cdot\left(\sin2x-\cos2x\right)=3\)
\(\Leftrightarrow\left(\sin2x-\cos2x\right)=0\)
\(\Leftrightarrow\sqrt{2}\cdot\sin\left(2x-\frac{\pi}{4}\right)=0\)
\(\Leftrightarrow\sin\left(2x-\frac{\pi}{4}\right)=\sin0\)
\(\Leftrightarrow2x-\frac{\pi}{4}=k\pi\)
\(\Leftrightarrow x=\frac{\pi}{8}+k\frac{\pi}{2}\left(k\in Z\right)\)
\(2.\cos\frac{x}{2}+2\cdot\cos^2\left(2x-\frac{\pi}{8}\right)=1\)
\(\Leftrightarrow\cos\frac{x}{2}+2\cdot\cos^2\left(2x-\frac{\pi}{8}\right)-1=0\)
\(\Leftrightarrow\cos\frac{x}{2}+\cos\left(4x-\frac{\pi}{4}\right)=0\)
\(\Leftrightarrow2\cdot\cos\left(\frac{9x}{4}-\frac{\pi}{8}\right)\cdot\cos\left(\frac{-7x}{4}+\frac{\pi}{8}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}\cos\left(\frac{9x}{4}-\frac{\pi}{8}\right)=0\\\cos\left(\frac{-7x}{4}+\frac{\pi}{8}\right)=0\end{cases}}\)\(\Leftrightarrow\orbr{\begin{cases}x=\frac{5\pi}{18}+k\frac{4\pi}{9}\\x=-\frac{3\pi}{14}+k\frac{4\pi}{7}\end{cases}\left(k\in Z\right)}\)
\(3.\sin3x+1=2\cdot\sin^2\left(\frac{x}{2}+\frac{\pi}{6}\right)\)
\(\Leftrightarrow\cos\left(\frac{\pi}{2}-3x\right)+1-2\cdot\sin^2\left(\frac{x}{2}+\frac{\pi}{6}\right)=0\)
\(\Leftrightarrow\cos\left(\frac{\pi}{2}-3x\right)+\cos\left(x+\frac{\pi}{3}\right)=0\)
\(\Leftrightarrow2\cdot\cos\left(-x+\frac{5\pi}{12}\right)\cdot\cos\left(-2x+\frac{\pi}{12}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{-\pi}{12}+k\pi\\x=-\frac{5\pi}{24}+k\frac{\pi}{2}\end{cases}}\left(k\in Z\right)\)
\(4.\sin^42x+\cos^42x=\frac{5}{8}\)
\(\Leftrightarrow\frac{\left(1-\cos4x\right)^2}{4}+\frac{\left(1+\cos4x\right)^2}{4}=\frac{5}{8}\)
\(\Leftrightarrow2\cdot\cos^24x+2=\frac{5}{2}\)
\(\Leftrightarrow\cos^24x=\frac{1}{4}\)
\(\Leftrightarrow\orbr{\begin{cases}\cos4x=\frac{1}{2}\\\cos4x=\frac{-1}{2}\end{cases}}\)\(\Leftrightarrow\orbr{\begin{cases}x=\pm\frac{\pi}{12}+k\frac{\pi}{2}\\x=\pm\frac{2\pi}{3}+k\frac{\pi}{2}\end{cases}}\left(k\in Z\right)\)
\(5.\cos\left(3x-\frac{\pi}{6}\right)-\cos\left(3x+\frac{\pi}{6}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow-2\cdot\sin3x\cdot\sin\left(\frac{-\pi}{6}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\sin3x=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{\pi}{12}+k\frac{2\pi}{3}\\x=\frac{\pi}{4}+k\frac{2\pi}{3}\end{cases}\left(k\in Z\right)}\)
dfgxdfsfddddddffffffffffffffffffffffffds xcvvvvvvvvc
555555555555555555
999999999999999999999999999999+9999999999999999999999999999=1999999999999999999999999999999999998 nhé
