- LG a
- LG b
Đưa các biểu thức sau về dạng \[C\sin[x + α]\]
LG a
\[\sin x + \tan {\pi \over 7}\cos x\]
Lời giải chi tiết:
Ta có:
\[\begin{array}{l}
\sin x + \tan \frac{\pi }{7}\cos x\\
= \sin x + \frac{{\sin \frac{\pi }{7}}}{{\cos \frac{\pi }{7}}}.\cos x\\
= \sin x + \frac{{\sin \frac{\pi }{7}\cos x}}{{\cos \frac{\pi }{7}}}\\
= \frac{{\sin x\cos \frac{\pi }{7} + \sin \frac{\pi }{7}\cos x}}{{\cos \frac{\pi }{7}}}\\
= \frac{{\sin \left[ {x + \frac{\pi }{7}} \right]}}{{\cos \frac{\pi }{7}}}
\end{array}\]
\[ = \frac{1}{{\cos \frac{\pi }{7}}}\sin \left[ {x + \frac{\pi }{7}} \right]\]
LG b
\[\tan {\pi \over 7}\sin x + \cos x\]
Lời giải chi tiết:
\[\begin{array}{l}
\tan \frac{\pi }{7}\sin x + \cos x\\
= \frac{{\sin \frac{\pi }{7}}}{{\cos \frac{\pi }{7}}}.\sin x + \cos x\\
= \frac{{\sin \frac{\pi }{7}\sin x}}{{\cos \frac{\pi }{7}}} + \cos x\\
= \frac{{\sin \frac{\pi }{7}\sin x + \cos x\cos \frac{\pi }{7}}}{{\cos \frac{\pi }{7}}}\\
= \frac{{\cos \left[ {x - \frac{\pi }{7}} \right]}}{{\cos \frac{\pi }{7}}} = \frac{{\cos \left[ {\frac{\pi }{7} - x} \right]}}{{\cos \frac{\pi }{7}}}\\
= \frac{{\sin \left[ {\frac{\pi }{2} - \frac{\pi }{7} + x} \right]}}{{\cos \frac{\pi }{7}}}\\
= \frac{{\sin \left[ {\frac{{5\pi }}{{14}} + x} \right]}}{{\cos \frac{\pi }{7}}}\\
= \frac{1}{{\cos \frac{\pi }{7}}}\sin \left[ {x + \frac{{5\pi }}{{14}}} \right]
\end{array}\]