continuity of sine and cosine
ContinuityOfSineAndCosine
2007-04-25 13:21:52
2007-09-22 14:15:29
Theorem
continuous
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\textbf{Theorem.}\, The real functions \;$x\mapsto\sin{x}$\;
and\; $x\mapsto\cos{x}$\; are continuous at every real number $x$.
{\em Proof.}\, Let $\varepsilon$ be an arbitrary positive number.\,
Denote\, $\Delta\sin{x} := \sin{z}-\sin{x}$,\,
$\Delta\cos{x} := \cos{z}-\cos{x}$\, where we suppose that\,
$|z-x| < \frac{\pi}{2}$.\, We may interpret $|z-x|$ as an arc
of the unit circle of the $xy$-plane.\, Let's think in the
circle the right triangle with hypotenuse the chord of the arc and
the catheti (i.e. the shorter sides) vertical and horizontal.\, Then
$|\Delta\sin{x}|$ and $|\Delta\cos{x}|$ are just these cathets; so we have
$$|\Delta\sin{x}| \leqq |z-x|,\;\; |\Delta\cos{x}| \leqq |z-x|.$$
If we make\, $|z-x| < \varepsilon$,\, then also\, $|\Delta\sin{x}|$ and
$|\Delta\cos{x}|$ are less than $\varepsilon$.\, It means that both
functions are continuous at $x$.\\
\begin{figure}
\begin{center}
\includegraphics{circle.eps}
\end{center}
\caption{Geometric bounds on $\left| \Delta \cos x \right|$ and $\left| \Delta \sin x \right|$}
\end{figure}
\begin{thebibliography}{9}
\bibitem{NP}{\sc E. Lindel\"of:} {\em Johdatus korkeampaan analyysiin}. Nelj\"as painos.\, Werner S\"oderstr\"om Osakeyhti\"o, Porvoo ja Helsinki (1956).
\end{thebibliography}