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 $\epsilon$ be an arbitrary positive number. Denote  $\mathrm{\Delta }\sin x=:\sin z-\sin x$, $\mathrm{\Delta }\cos x=:\cos z-\cos x$  where we suppose that .  We may interpret $|z-x|$ as an arcof the unit circle of the $xy$-plane.  Let’s think in thecircle the right triangle with hypotenuse the chord of the arc andthe catheti (i.e. the shorter sides) vertical and horizontal.  Then$|\mathrm{\Delta }\sin x|$ and $|\mathrm{\Delta }\cos x|$ are just these cathets; so we have

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

If we make  ,  then also  $|\mathrm{\Delta }\sin x|$ and$|\mathrm{\Delta }\cos x|$ are less than $\epsilon$.  It means that bothfunctions are continuous at $x$.