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Giải các phương trình sau
a) \(3{\cos ^2}x – 2\sin x + 2 = 0\)
b) \(5{\sin ^2}x + 3\cos x + 3 = 0\)
c) \({\sin ^6}x + {\cos ^6}x = 4{\cos ^2}2x\)
d) \( – {1 \over 4} + {\sin ^2}x = {\cos ^4}x\)
a)
\(\eqalign{
& 3{\cos ^2}x – 2\sin x + 2 = 0 \cr
& \Leftrightarrow 3\left( {1 – {{\sin }^2}x} \right) – 2\sin x + 2 = 0 \cr
& \Leftrightarrow 3{\sin ^2}x + 2\sin x – 5 = 0 \cr
& \Leftrightarrow \left( {\sin x – 1} \right)\left( {3\sin x + 5} \right) = 0 \cr
& \Leftrightarrow \sin x = 1 \cr
& \Leftrightarrow x = {\pi \over 2} + k2\pi ,k \in {\rm Z} \cr} \)
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b)
\(\eqalign{
& 5{\sin ^2}x + 3\cos x + 3 = 0 \cr
& \Leftrightarrow 5\left( {1 – {{\cos }^2}x} \right) + 3\cos x + 3 = 0 \cr
& \Leftrightarrow 5{\cos ^2}x – 3\cos x – 8 = 0 \cr
& \Leftrightarrow \left( {\cos x + 1} \right)\left( {5\cos x – 8} \right) = 0 \cr
& \Leftrightarrow \cos x = – 1 \cr
& \Leftrightarrow x = \left( {2k + 1} \right)\pi ,k \in {\rm Z} \cr} \)
c)
\(\eqalign{
& {\sin ^6}x + {\cos ^6}x = 4{\cos ^2}2x \cr
& \Leftrightarrow {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^3} – 3{\sin ^2}x{\cos ^2}x\left( {{{\sin }^2}x + {{\cos }^2}x} \right) = 4{\cos ^2}2x \cr
& \Leftrightarrow 1 – {3 \over 4}{\sin ^2}2x = 4{\cos ^2}2x \cr
& \Leftrightarrow 1 – {3 \over 4}\left( {1 – {{\cos }^2}2x} \right) = 4{\cos ^2}2x \cr
& \Leftrightarrow {{13} \over 4}{\cos ^2}2x = {1 \over 4} \cr
& \Leftrightarrow 13\left( {{{1 + \cos 4x} \over 2}} \right) = 1 \cr
& \Leftrightarrow 1 + \cos 4x = {2 \over {13}} \cr
& \Leftrightarrow \cos 4x = – {{11} \over {13}} \cr
& \Leftrightarrow 4x = \pm \arccos \left( { – {{11} \over {13}}} \right) + k2\pi ,k \in {\rm Z} \cr
& \Leftrightarrow x = \pm {1 \over 4}\arccos \left( { – {{11} \over {13}}} \right) + k{\pi \over 2},k \in {\rm Z} \cr} \)
d)
\(\eqalign{
& – {1 \over 4} + {\sin ^2}x = {\cos ^4}x \cr
& \Leftrightarrow – {1 \over 4} + {{1 – \cos 2x} \over 2} = {\left( {{{1 + \cos 2x} \over 2}} \right)^2} \cr
& \Leftrightarrow – 1 + 2 – 2\cos 2x = 1 + 2\cos 2x + {\cos ^2}2x \cr
& \Leftrightarrow {\cos ^2}2x + 4\cos 2x = 0 \cr
& \Leftrightarrow \left[ \matrix{
\cos 2x = 0 \hfill \cr
\cos 2x = – 4\left( {Vô\,\,nghiệm} \right){\rm{ }} \hfill \cr} \right. \cr
& \Leftrightarrow 2x = {\pi \over 2} + k\pi ,k \in {\rm Z} \cr
& \Leftrightarrow x = {\pi \over 4} + k{\pi \over 2},k \in {\rm Z} \cr} \)