Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

**Yuguang Yang**- Shinshu University, Nagano
**Yasunari Shidama**- Shinshu University, Nagano

- In this article, we defined {\em sinus} and {\em cosine} as the real part and the imaginary part of the exponential function on complex, and also give their series expression. Then we proved the differentiablity of {\em sinus}, {\em cosine} and the exponential function of real. Finally, we showed the existence of the circle ratio, and some formulas of {\em sinus}, {\em cosine}.

- Some Definitions and Properties of Complex Sequence
- Definition of Exponential Function on Complex
- Definition of Sinus, Cosine, and Exponential Function on ${\Bbb R}$
- Differential of Sinus, Cosine, and Exponential Function
- Existence of Circle Ratio
- Formulas of Sinus, Cosine

- [1]
Agnieszka Banachowicz and Anna Winnicka.
Complex sequences.
*Journal of Formalized Mathematics*, 5, 1993. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
The complex numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Library Committee.
Introduction to arithmetic.
*Journal of Formalized Mathematics*, Addenda, 2003. - [8]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz.
Partial functions from a domain to the set of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Rafal Kwiatek.
Factorial and Newton coefficients.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Adam Naumowicz.
Conjugate sequences, bounded complex sequences and convergent complex sequences.
*Journal of Formalized Mathematics*, 8, 1996. - [15]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [16]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Konrad Raczkowski.
Integer and rational exponents.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Konrad Raczkowski and Andrzej Nedzusiak.
Series.
*Journal of Formalized Mathematics*, 3, 1991. - [19]
Konrad Raczkowski and Pawel Sadowski.
Real function continuity.
*Journal of Formalized Mathematics*, 2, 1990. - [20]
Konrad Raczkowski and Pawel Sadowski.
Real function differentiability.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [22]
Yasunari Shidama and Artur Kornilowicz.
Convergence and the limit of complex sequences. Series.
*Journal of Formalized Mathematics*, 9, 1997. - [23]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [24]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [25]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

[ Download a postscript version, MML identifier index, Mizar home page]