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[parent] continuity of sine and cosine (Theorem)

Theorem. The real functions $ x\mapsto\sin{x}$ and $ x\mapsto\cos{x}$ are continuous at every real number $ x$.

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 $ \vert z-x\vert < \frac{\pi}{2}$. We may interpret $ \vert z-x\vert$ 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 $ \vert\Delta\sin{x}\vert$ and $ \vert\Delta\cos{x}\vert$ are just these cathets; so we have

$\displaystyle \vert\Delta\sin{x}\vert \leqq \vert z-x\vert,\;\; \vert\Delta\cos{x}\vert \leqq \vert z-x\vert.$
If we make $ \vert z-x\vert < \varepsilon$, then also $ \vert\Delta\sin{x}\vert$ and $ \vert\Delta\cos{x}\vert$ are less than $ \varepsilon$. It means that both functions are continuous at $ x$.
Figure: Geometric bounds on $ \left\vert \Delta \cos x \right\vert$ and $ \left\vert \Delta \sin x \right\vert$
\includegraphics{circle.eps}

Bibliography

1
E. LINDELÖF: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).



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Cross-references: functions, sides, catheti, chord, hypotenuse, right triangle, circle, unit circle, arc, positive, real number, continuous at, real functions
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This is version 7 of continuity of sine and cosine, born on 2007-04-25, modified 2007-09-22.
Object id is 9263, canonical name is ContinuityOfSineAndCosine.
Accessed 997 times total.

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AMS MSC: 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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