continuity of sine and cosine

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 $|z-x|<\frac{\pi}{2}$ . We may interpret $|z-x|$ as an arc of the unit circle of the $x y$ -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|,\quad|\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 $x$ .

References

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

Title	continuity of sine and cosine
Canonical name	ContinuityOfSineAndCosine
Date of creation	2013-03-22 19:15:37
Last modified on	2013-03-22 19:15:37
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	4
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26A15