Bài 5. Giải các phương trình sau:
a) \(cosx - \sqrt3sinx = \sqrt2\);
b) \(3sin3x - 4cos3x = 5\);
c) \(2sin2x + 2cos2x - \sqrt2 = 0\);
d) \(5cos2x + 12sin2x -13 = 0\).
a) \(cosx - \sqrt3sinx = \sqrt2\)
\( \Leftrightarrow {1 \over 2}\cos x - {{\sqrt 3 } \over 2}{\mathop{\rm sinx}\nolimits} = {{\sqrt 2 } \over 2}\)
\( \Leftrightarrow \cos x.\cos {\pi \over 3} - \sin x\sin {\pi \over 3} = \cos {\pi \over 4}\)
\( \Leftrightarrow \cos \left( {x + {\pi \over 3}} \right) = \cos {\pi \over 4}\)
\( \Leftrightarrow \left[ \matrix{
x + {\pi \over 3} = {\pi \over 4} + k2\pi \hfill \cr
x + {\pi \over 3} = - {\pi \over 4} + k2\pi \hfill \cr} \right.\)
\( \Leftrightarrow \left[ \matrix{
x = - {\pi \over {12}} + k2\pi \hfill \cr
x = - {{7\pi } \over {12}} + k2\pi \hfill \cr} \right.(k \in\mathbb{Z} )\)
b) \(3sin3x - 4cos3x = 5\)
\( \Leftrightarrow {3 \over 5}\sin 3x - {4 \over 5}\cos 3x = 1\)
Đặt \(\alpha =arccos{3\over5}\) thì phương trình trở thành
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\(cosαsin3x - sinαcos3x = 1\)\( ⇔ sin(3x - α) = 1\)
\( ⇔ 3x - α = {\pi\over2} + k2π\)
\( \Leftrightarrow x = {\pi \over 6} + {\alpha \over 3} + {{k2\pi } \over 3}(k \in \mathbb{Z})\)
c) \(2sin2x + 2cos2x - \sqrt2 = 0\)
\(\Leftrightarrow {1 \over {\sqrt 2 }}\sin 2x + {1 \over {\sqrt 2 }}\cos 2x = {1 \over 2}\)
\( \Leftrightarrow \sin 2x.\cos {\pi \over 4} + \cos 2x.\sin {\pi \over 4} = \sin {\pi \over 6}\)
\( \Leftrightarrow \sin \left( {2x + {\pi \over 4}} \right) = \sin {\pi \over 6}\)
\( \Leftrightarrow \left[ \matrix{
2x + {\pi \over 4} = {\pi \over 6} + k2\pi \hfill \cr
2x + {\pi \over 4} = \pi - {\pi \over 6} + k2\pi \hfill \cr} \right.\)
\( \Leftrightarrow \left[ \matrix{
x = - {\pi \over {12}} + k\pi \hfill \cr
x = {{7\pi } \over {12}} + k\pi \hfill \cr} \right.(k \in \mathbb{Z})\)
d) \(5cos2x + 12sin2x -13 = 0\)
\( \Leftrightarrow {5 \over {13}}\cos 2x + {{12} \over {13}}\sin 2x = 1\)
Đặt \(\alpha = arccos{5\over13}\) thì phương trình trở thành
\(cosαcos2x + sinαsin2x = 1 ⇔ cos(2x - α) = 1\)
\(⇔ 2x-\alpha = k2π\) \(\Leftrightarrow x={\alpha\over2}+k\pi\), \((k ∈ \mathbb{Z})\)
(trong đó \(α = arccos{5\over13})\).