Bài tập 4 trang 223 SGK Toán 11 NC

Bài tập 4 trang 223 SGK Toán 11 NC

a.

\(\begin{array}{l}
{\sin ^4}x + {\cos ^4}x = \frac{3}{4}\\
 \Leftrightarrow 1 - 2{\sin ^2}x{\cos ^2}x = \frac{3}{4}\\
 \Leftrightarrow 1 - \frac{1}{2}{\sin ^2}2x = \frac{3}{4}\\
 \Leftrightarrow {\sin ^2}2x = \frac{1}{2} \Leftrightarrow \frac{{1 - \cos 4x}}{2} = \frac{1}{2}\\
 \Leftrightarrow \cos 4x = 0 \Leftrightarrow x = \frac{\pi }{8} + k\frac{\pi }{4},k \in Z
\end{array}\)

b.

\(\begin{array}{l}
{\sin ^2}2x - {\sin ^2}x = {\sin ^2}\frac{\pi }{4}\\
 \Leftrightarrow 4{\sin ^2}x{\cos ^2}x - {\sin ^2}x = \frac{1}{2}\\
 \Leftrightarrow 8{\sin ^2}x\left( {1 - {{\sin }^2}x} \right) - 2{\sin ^2}x = 1\\
 \Leftrightarrow 8{\sin ^4}x - 6{\sin ^2}x + 1 = 0\\
 \Leftrightarrow \left[ \begin{array}{l}
{\sin ^2}x = \frac{1}{2}\\
{\sin ^2}x = \frac{1}{4}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\frac{{1 - \cos 2x}}{2} = \frac{1}{2}\\
\frac{{1 - \cos 2x}}{2} = \frac{1}{4}
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
\cos 2x = \frac{1}{2}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{4} + k\frac{\pi }{2}\\
x = \frac{\pi }{6} + k\frac{\pi }{2}
\end{array} \right.
\end{array}\)

c.

\(\begin{array}{l}
\cos x\cos 2x = \cos 3x\\
 \Leftrightarrow \frac{1}{2}\left( {\cos 3x + \cos x} \right) = \cos 3x\\
 \Leftrightarrow \cos 3x = \cos x\\
 \Leftrightarrow \left[ \begin{array}{l}
3x = x + k2\pi \\
3x =  - x + k2\pi 
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = k\pi \\
x = k\frac{\pi }{2}
\end{array} \right.\\
 \Leftrightarrow x = k\frac{\pi }{2},k \in Z
\end{array}\)

d. Điều kiên: \(\cos 2x \ne 0\)

\(\begin{array}{l}
\tan 2x - \sin 2x + \cos 2x - 1 = 0\\
 \Leftrightarrow \tan 2x(1 - \cos 2x) - (1 - \cos 2x) = 0\\
 \Leftrightarrow (1 - \cos 2x)(\tan 2x - 1) = 0\\
 \Leftrightarrow \left[ \begin{array}{l}
\tan 2x = 1\\
\cos 2x = 1
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{8} + k\frac{\pi }{2}\\
x = k\pi 
\end{array} \right.,k \in Z
\end{array}\)

-- Mod Toán 11