Giải các phương trình sau:

a) \(2\sin \left( {\frac{x}{3} + 15^\circ } \right) + \sqrt 2 = 0\);

b) \(\cos \left( {2x + \frac{\pi }{5}} \right) = - 1\);

c) 3tan 2x + \(\sqrt 3 \) = 0;

d) cot (2x – 3) = cot 15°.

Lời giải

a) \(2\sin \left( {\frac{x}{3} + 15^\circ } \right) + \sqrt 2 = 0\)

\( \Leftrightarrow \sin \left( {\frac{x}{3} + 15^\circ } \right) = - \frac{{\sqrt 2 }}{2}\)

\( \Leftrightarrow \sin \left( {\frac{x}{3} + 15^\circ } \right) = \sin \left( { - 45^\circ } \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}\frac{x}{3} + 15^\circ = - 45^\circ + k360^\circ \\\frac{x}{3} + 15^\circ = 180^\circ - \left( { - 45^\circ } \right) + k360^\circ \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}x = - 180^\circ + k1080^\circ \\x = 630^\circ + k1080^\circ \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).

b) \(\cos \left( {2x + \frac{\pi }{5}} \right) = - 1\)

\( \Leftrightarrow 2x + \frac{\pi }{5} = \pi + k2\pi \,\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow x = \frac{{2\pi }}{5} + k\pi \,\,\left( {k \in \mathbb{Z}} \right)\).

c) 3tan 2x + \(\sqrt 3 \) = 0

\( \Leftrightarrow \tan 2x =  - \frac{{\sqrt 3 }}{3}\)

\( \Leftrightarrow \tan 2x = \tan \left( { - \frac{\pi }{6}} \right)\)

\( \Leftrightarrow 2x = - \frac{\pi }{6} + k\pi \,\,\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow x = - \frac{\pi }{{12}} + k\frac{\pi }{2}\,\,\left( {k \in \mathbb{Z}} \right)\).

d) cot (2x – 3) = cot 15°

⇔ 2x – 3 = 15° + k180° (k ∈ ℤ)

⇔ 2x = 3 + 15° + k180° (k ∈ ℤ)

⇔ x = 1,5 + 7,5° + k90° (k ∈ ℤ).