continuity of sine and cosine
Proof. Let be an arbitrary positive number. Denote , where we suppose that . We may interpret as an arc of the unit circle of the -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 and are just these cathets; so we have
If we make , then also and are less than . It means that both functions are continuous at .
- 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).