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Câu hỏi: Chứng minh rằng, với mọi \(\alpha \), với mọi số nguyên k, ta có:
\(\sin \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}{\left({ - 1} \right)^l}\sin \alpha nếu k - 2l\\{\left({ - 1} \right)^l}\cos \alpha nếu k = 2l + 1;\end{array} \right.\)
\(\cos \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}{\left({ - 1} \right)^l}\cos \alpha nếu k = 2l\\{\left({ - 1} \right)^{l + 1}}\sin \alpha nếu k = 2l + 1;\end{array} \right.\)
\(\tan \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}\tan \alpha nếu k = 2l + 1\\ - \cot \alpha nếu k = 2l + 1 \end{array} \right.\)
(khi các biểu thức này có nghĩa)
\(\sin \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}{\left({ - 1} \right)^l}\sin \alpha nếu k - 2l\\{\left({ - 1} \right)^l}\cos \alpha nếu k = 2l + 1;\end{array} \right.\)
\(\cos \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}{\left({ - 1} \right)^l}\cos \alpha nếu k = 2l\\{\left({ - 1} \right)^{l + 1}}\sin \alpha nếu k = 2l + 1;\end{array} \right.\)
\(\tan \left( {\alpha + k\dfrac{\pi }{2}} \right) = \left\{ \begin{array}{l}\tan \alpha nếu k = 2l + 1\\ - \cot \alpha nếu k = 2l + 1 \end{array} \right.\)
(khi các biểu thức này có nghĩa)
Lời giải chi tiết
• \(\sin \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \sin \left({\alpha + l\pi } \right) = {\left({ - 1} \right)^l}\sin \alpha \);
\(\begin{array}{l}\sin \left[ {\alpha + \left( {2l + 1} \right)\dfrac{\pi }{2}} \right] = \sin \left({\alpha + \dfrac{\pi }{2} + l\pi } \right)\\ = {\left({ - 1} \right)^l}\sin \left({\alpha + \dfrac{\pi }{2}} \right) = {\left({ - 1} \right)^l}\cos \alpha .\end{array}\)
• \(\begin{array}{l}\cos \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \cos \left({\alpha + l\pi } \right) = {\left({ - 1} \right)^l}\cos \alpha \\\cos \left[ {\alpha + \left({2l + 1} \right)\dfrac{\pi }{2}} \right] = \cos \left({\alpha + \dfrac{\pi }{2} + l\pi } \right)\\ = {\left({ - 1} \right)^l}\cos \left({\alpha + \dfrac{\pi }{2}} \right) = {\left({ - 1} \right)^l}\left({ - \sin \alpha } \right)\\ = {\left({ - 1} \right)^{l + 1}}\sin \alpha \end{array}\)
• Từ đó
\(\begin{array}{l}\tan \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \tan \alpha ;\\\tan \left[ {\alpha + \left({2l + 1} \right)\dfrac{\pi }{2}} \right] = - \cot \alpha .\end{array}\)
• \(\sin \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \sin \left({\alpha + l\pi } \right) = {\left({ - 1} \right)^l}\sin \alpha \);
\(\begin{array}{l}\sin \left[ {\alpha + \left( {2l + 1} \right)\dfrac{\pi }{2}} \right] = \sin \left({\alpha + \dfrac{\pi }{2} + l\pi } \right)\\ = {\left({ - 1} \right)^l}\sin \left({\alpha + \dfrac{\pi }{2}} \right) = {\left({ - 1} \right)^l}\cos \alpha .\end{array}\)
• \(\begin{array}{l}\cos \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \cos \left({\alpha + l\pi } \right) = {\left({ - 1} \right)^l}\cos \alpha \\\cos \left[ {\alpha + \left({2l + 1} \right)\dfrac{\pi }{2}} \right] = \cos \left({\alpha + \dfrac{\pi }{2} + l\pi } \right)\\ = {\left({ - 1} \right)^l}\cos \left({\alpha + \dfrac{\pi }{2}} \right) = {\left({ - 1} \right)^l}\left({ - \sin \alpha } \right)\\ = {\left({ - 1} \right)^{l + 1}}\sin \alpha \end{array}\)
• Từ đó
\(\begin{array}{l}\tan \left( {\alpha + 2l\dfrac{\pi }{2}} \right) = \tan \alpha ;\\\tan \left[ {\alpha + \left({2l + 1} \right)\dfrac{\pi }{2}} \right] = - \cot \alpha .\end{array}\)