# continuity of sine and cosine

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 $xy$-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

 $|\Delta\sin{x}|\;\leqq\;|z-x|,\quad|\Delta\cos{x}|\;\leqq\;|z-x|.$

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 ContinuityOfSineAndCosine 2013-03-22 19:15:37 2013-03-22 19:15:37 pahio (2872) pahio (2872) 4 pahio (2872) Theorem msc 26A15