Giải các phương trình:
a) \(\cos (x + 3) = \frac{1}{3}\)
b) \(\cos (3x - {45^o}) = \frac{{\sqrt 3 }}{2}\)
c) \(\cos (2x + \frac{\pi }{3}) = - \frac{1}{2}\)
d) \((2 + \cos x)(3\cos 2x - 1) = 0\)
a) \(\cos (x + 3) = \frac{1}{3}\)
\( \Leftrightarrow x + 3 = \pm \arccos 13 + k2\pi ,k \in Z\)
\( \Leftrightarrow x = - 3 \pm \arccos \frac{1}{3} + k2\pi ,k \in Z\)
b) \(\cos (3x - {45^o}) = \frac{{\sqrt 3 }}{2}\)
⇔ cos(3x−45o) = cos30o
\( \Leftrightarrow 3x - {45^o} = \pm {30^o} + k{360^o},k \in Z\)
\( \Leftrightarrow \left[ \begin{array}{l}
x = {25^o} + k{120^o},k \in Z\\
x = {5^o} + k{120^o},k \in Z
\end{array} \right.\)
c) \(\cos (2x + \frac{\pi }{3}) = - \frac{1}{2}\)
\( \Leftrightarrow \cos (2x + \frac{\pi }{3}) = \cos \left( {\frac{{2\pi }}{3}} \right)\)
\( \Leftrightarrow 2x + \frac{\pi }{3} = \pm \frac{{2\pi }}{3} + k2\pi ,k \in Z\)
\( \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{6} + k\pi ,k \in Z\\
x = - \frac{\pi }{2} + k\pi ,k \in Z
\end{array} \right.\)
d) (2+cosx)(3cos2x−1) = 0
\( \Leftrightarrow \left[ {\begin{array}{*{20}{l}}
{2 + \cos x = 0}\\
{3\cos 2x - 1 = 0}
\end{array}} \right.\)
Nếu cosx = −2 (vô nghiệm)
Nếu \(\cos 2x = \frac{1}{3}\)
\( \Leftrightarrow 2x = \pm \arccos \frac{1}{3} + k2\pi ,k \in Z\)
\( \Leftrightarrow x = \pm 12\arccos \frac{1}{3} + k\pi ,k \in Z\)
-- Mod Toán 11