Bài 3.3 trang 36 Sách bài tập (SBT) Đại số và giải tích 11

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) \( - {1 \over 4} + {\sin ^2}x = {\cos ^4}x\)

Giải: 

a)

\(\eqalign{
& 3{\cos ^2}x - 2\sin x + 2 = 0 \cr
& \Leftrightarrow 3\left( {1 - {{\sin }^2}x} \right) - 2\sin x + 2 = 0 \cr
& \Leftrightarrow 3{\sin ^2}x + 2\sin x - 5 = 0 \cr
& \Leftrightarrow \left( {\sin x - 1} \right)\left( {3\sin x + 5} \right) = 0 \cr
& \Leftrightarrow \sin x = 1 \cr
& \Leftrightarrow x = {\pi \over 2} + k2\pi ,k \in {\rm Z} \cr} \)

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) 

\(\eqalign{
& - {1 \over 4} + {\sin ^2}x = {\cos ^4}x \cr
& \Leftrightarrow - {1 \over 4} + {{1 - \cos 2x} \over 2} = {\left( {{{1 + \cos 2x} \over 2}} \right)^2} \cr
& \Leftrightarrow - 1 + 2 - 2\cos 2x = 1 + 2\cos 2x + {\cos ^2}2x \cr
& \Leftrightarrow {\cos ^2}2x + 4\cos 2x = 0 \cr
& \Leftrightarrow \left[ \matrix{
\cos 2x = 0 \hfill \cr
\cos 2x = - 4\left( {Vô\,\,nghiệm} \right){\rm{ }} \hfill \cr} \right. \cr
& \Leftrightarrow 2x = {\pi \over 2} + k\pi ,k \in {\rm Z} \cr
& \Leftrightarrow x = {\pi \over 4} + k{\pi \over 2},k \in {\rm Z} \cr} \)