The Paratrigonometry. Nonconventional trigonometry.

CONTENTS

1. Elements of Polygonal Trigonometry 11
2. Regarding the general character of the Polygonal Trigonometry 16
3. Periodic transtrigonometric functions 22
4. The Infratrigonometry, an inferior order neighborhood domain of the Transtrigonometry 27
5. The Ultratrigonometry, a superior order adjacent domain of the Transtrigonometry 32
6. The Paratrigonometry and some of its specific symmetries 38
7. Regarding the paratrigonometric equation of the circle 48
8. Some developments of fundamental trigonometric equation 53
9. The application of the paratrigonometric fundamental equations in the four trigonometric quadrants 58
10. An application case of the paratrigonometric polar coordinates 64
11. Paratrigonometric functions raised to some powers and their applications 70
12. Paratrigonometric functions relative to the finite spirals as the Basic Trigonometric Figures 79
13. Possibilities to represent some symmetric functions by unified equations and some of their applications in the Paratrigonometry 87
14. A simplified alternative to develop periodic functions in Trigonometric Fourier Series 97
References 103

ABOUT AUTHORS

BIOGRAPHY OF MALVINA BAICA
The wife of Adrian Baica and daughter of Adam (a lawyer) and Cornelia Bunghiu, Malvina Florica Baica was born in Oravita, Romania, on the 3rd of November 1942; After gaining her baccalaureate at Oravita Lyceum in 1960 she spent, five years at the University of Timisoara, earning a BS in Mathematics and Physics (mechanics) and an European MS in Mathematics (Projective and Differential Geometry). She taught for two years, but subsequently emigrated, with political asylum to the United States. Resuming her studies she was awarded her MA in Mathematics (Algebra and Number Theory) by Illinois Institute of Technology of Chicago in 1974 and her Ph.D. in Mathematics (Algebraic Number Theory and Algebra) by the University of Houston, Texas, in 1980. Her Ph.D. advisor was Jurgen Schmidt and her Ph.D. dissertation advisors were Helmut Hasse and Leon Bernstein.
Since 1992, Professor of Mathematics and Computer Science at the University of Wisconsin – Whitewater, Dr. Baica jointed the academic staff there in 1984 as a lecturer, and was appointed assistant professor the following year. She had previously held positions as assistant professor at Marquette University (Wisconsin), Marshall University (West Virginia) and Valparaiso University (Indiana).
Her research results were published by reputable journals in more than 30 papers.
Among her publications the most significant are “An Algorithm in a Complex Field (ACF) and its Applications to the Calculations of Units” (which is a new and very powerful algorithm, which turned out to be the General Euclidean Algorithm) known as Baica’s General Euclidean Algorithm (BGEA); “Trigonometric Identities” (now known in mathematical literature as Baica’s Trigonometric Identities), where Gauss’ two trigonometric identities given by Gauss without proof become particular cases of Baica’s Trigonometric Identities proved true in general; “Putting last Digits first yields Multiples” (a joint paper with the late Helmut Hasse). Using her (BGEA) algorithm she proved all the famous opes problems in algebraic number theory, including Fermat Last Theorem (FLT).
She has given professional talks at Conferences and other meetings including the International Congress of Mathematicians, held at the University of California, Berkley, in 1986, the International Conference of Number Theory in Quebec, Canada, in 1987, the American Mathematical Society’s Centennial Celebration in Providence, Rhode Island, in 1988 and the International Conferences on Analytical Number Theory at the University of Illinois, Allerton Park, Urbana, also in 1988. After 1989 (Romanian Revolution) she returned to Alma Mater giving more conferences on her favorite subjects in Romania. As consequence she joined Pannonian Applied Mathematical Meetings (PAMM), finding a favorable climate to meet prestigious European mathematicians and scientists for mathematical and interdisciplinary cooperation.
She had received a distinguished professional leadership award and is included in the “International Directory of Distinguished Leadership”, third edition. Also, she is included in “Who is Who” International and in “Who is Who” for Women.

BIOGRAPHY OF MIRCEA CARDU
Born in Gradinari, Caras-Severin county – Romania, on the 31st October 1934. After gaining his baccalaureate at Oravita Lyceum General Dragalina in 1952 he spent five years (until 1957) at the University “Politehnica” of Timisoara earning a BS, as dipl. engineer, specialized in Thermic Machines in the frame of Mechanical Faculty.
In the period 1957 – 1958 he worked at the Automotive Reparation Company (IRA) No. 13 from Timisoara as foreman, and in the period 1959 – 1968 at the Machinebuilding Works from Resita (UCMR), Department of steam turbine design.
In the period 1968 – 1994 he worked at the Researchand Design Institute for Thermopower Equipment (ICPET) from Bucharest, as chief of Design Department for turbo-machines, and general manager, in the period 1979 – 1986 and 1990 – 1994 respectively.
Always was keen on Mathematics which make up the support of a lot of published scientific papers as, for example:
• “About the computer programming of the steam turbine calculus” (in Romanian), Energetica Review, No .5 – 6, 1978;
• “The optimizing of angles ?1 of impulse steam turbines nozzles”, Revue Roumain de Sciences Techniques – Serie Electrotechnique et Energetique, 31, No. 2, 1986;
• “Regarding a global methodology to estimate the energy-ecologic efficiency of thermopower plants” – co-author: M. Baica, Energy Conversion and Management, Vol. 40, No. 1, 1999.
He was specialized in the field of steam turbines and other thermopower equipment at the following famous companies: LMZ Sankt Petersburg from Russia, Rateau-Schneider and Alstom from France, Clarck and General Electric from USA, ENEL from Italy (at the AEIA proposal), AECL from Canada.
With the business purposes (export, technical co-operation etc.) he was also in: Germany, Hungary, Poland, Czechoslovakia, Ukraine, Bulgaria, Greece, China, India, Sri-Lanka.
His research results were published by journals from Romania and USA in more than 100 papers. Also, he is the author of 18 conferences and 8 patented inventions.
He is the author of the book “Paths to the all horizons (From an engineer diary)”, AGIR Publishing House, Bucharest, 1996. Also, is the co-author of the book “Steam turbines”, Technical Publishing House, Bucharest, 1976.
He had received the following Romanian State medals: “Work Order”, “Scientific Merit Order”, “Order Tudor Vladimirescu”.
He is member of the “General Association of the Engineers in Romania (AGIR)”, the “Romanian National Comitee of the World Energy Council” and of the “Romanian Institute of Energy (IRE)”.