Bài tập 40 trang 213 SGK Toán 10 NC
Bài tập 40 trang 213 SGK Toán 10 NC
Chứng minh rằng:
\(\begin{array}{*{20}{l}}
{a){\mkern 1mu} {\mkern 1mu} \sin \alpha + \cos \alpha = \sqrt 2 \sin \left( {\alpha + \frac{\pi }{4}} \right)}\\
{b){\mkern 1mu} {\mkern 1mu} \sin \alpha - \cos \alpha = \sqrt 2 \sin \left( {\alpha - \frac{\pi }{4}} \right)}\\
\begin{array}{l}
c){\mkern 1mu} {\mkern 1mu} \tan \left( {\frac{\pi }{4} - \alpha } \right) = \frac{{1 - \tan \alpha }}{{1 + \tan \alpha }}\\
\left( {\alpha \ne \frac{\pi }{2} + k\pi ;\alpha \ne \frac{{3\pi }}{4} + k\pi } \right)
\end{array}\\
\begin{array}{l}
d){\mkern 1mu} {\mkern 1mu} \tan \left( {\frac{\pi }{4} + \alpha } \right) = \frac{{1 + \tan \alpha }}{{1 - \tan \alpha }}\\
\left( {\alpha \ne \frac{\pi }{2} + k\pi ;\alpha \ne \frac{\pi }{4} + k\pi } \right)
\end{array}
\end{array}\)
a)
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\sqrt 2 \sin \left( {\alpha + \frac{\pi }{4}} \right)\\
= \sqrt 2 \left( {\sin \alpha .\cos \frac{\pi }{4} + \sin \frac{\pi }{4}.\cos \alpha } \right)
\end{array}\\
\begin{array}{l}
= \sqrt 2 \left( {\sin \alpha .\frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}\cos \alpha } \right)\\
= \sin \alpha + \cos \alpha
\end{array}
\end{array}\)
b)
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\sqrt 2 \sin \left( {\alpha - \frac{\pi }{4}} \right)\\
= \sqrt 2 \left( {\sin \alpha .\cos \frac{\pi }{4} - \sin \frac{\pi }{4}.\cos \alpha } \right)
\end{array}\\
\begin{array}{l}
= \sqrt 2 \left( {\sin \alpha .\frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2}\cos \alpha } \right)\\
= \sin \alpha - \cos \alpha
\end{array}
\end{array}\)
c)
\(\begin{array}{l}
\tan \left( {\frac{\pi }{4} - \alpha } \right) = \frac{{\tan \frac{\pi }{4} - \tan \alpha }}{{1 + \tan \frac{\pi }{4}.\tan \alpha }}\\
= \frac{{1 - \tan \alpha }}{{1 + \tan \alpha }}
\end{array}\)
d)
\(\begin{array}{l}
\tan \left( {\frac{\pi }{4} + \alpha } \right) = \frac{{\tan \frac{\pi }{4} + \tan \alpha }}{{1 - \tan \frac{\pi }{4}.\tan \alpha }}\\
= \frac{{1 + \tan \alpha }}{{1 - \tan \alpha }}
\end{array}\)
-- Mod Toán 10
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