#### Foreword

To many generations trigonometry was one of the mathematical
subjects taught in school, along with arithmetic, algebra, geometry, and
analysis. For some time now, due to the need to diminish the syllabus,
trigonometry stops being considered a distinct subject with its own
handbook, and is included in the syllabus for algebra, geometry and
analysis. Under these circumstances, the unitary nature of trigonometry is
sacrificed. But there are still enough pupils interested in mathematics,
enough graduates and students, not to mention the math teachers, the
numerous engineers and economists that are still fond of elementary,
school mathematics, but who would like to reconsider it from a superior
point of view, that is from the perspective of mathematic knowledge
accumulated in time. This book, written rigorously, but passionately, has a
perspective that is equally based on the ensemblist structure of modern
mathematics, on the elements of real analysis and the axiomatic
fundaments of geometry, and it reaches climax with the applications that
imply analytic algebraic and geometric abilities. From the historical
presentation of mathematics given in the book the reader can, for
example, learn the surprising fact that, from a chronologically point of view,
plane trigonometry was preceded by spherical trigonometry, necessary in
astronomy, although the latter is more complex than the former. The
author of this book, Professor Vasile Postolică, teaching at “Vasile
Alecsandri” University of Bacău, is a mathematician that has convincingly
asserted himself in research, with results published in exacting
publications, results that have become a part of the international circuit.
His vast experience in school and university teaching made possible the
development of this successful experiment, experienced by few
professors: the re-writing of a chapter of school mathematics from a
superior point of view. This adventurous path has been taken by
mathematicians like Felix Klein and Jacques Hadamard, hence this is a
very serious project. We congratulate Professor Vasile Postolică for this
success. At the same time, supposing that he will continue this adventure,
we suggest that he broaden the perspective by introducing elements of
trigonometric series and Fourier series, that are, probably, the most
important use of the trigonometric functions, having a particular relevance
in physics. Some elements of spherical trigonometry would be also timely.
The present book deserves to be successful in book stores and justifies
every one of our praises to its author.

Academician Solomon Marcus,

Piatra NeamN,

România

#### Preface

Trigonometry appeared as a part of medieval geometry, at first
the spherical component required by astronomy, after that plane
trigonometry was developed. In 1260 the most extensive work in this field
was created “ The Treatise of the Complete Quadrilateral”, written by
Näsr-ad-Din, followed by the first synthesis of trigonometric knowledge of
the 15th century elaborated by Regiomontanus. By studying and
developing applied mathematics, the Indians have introduced the sine
function as association of any circle arc having half the length of the under
tightened cord by the arc’s double, the term “sine” and some interpolator
methods of calculating the values of this function were given by
Peuerbach, and the natural connections between algebra, mathematical
analysis and trigonometry appeared in the 18th century. Using the major
elements of the Basis of Mathematics, the book briefly, but rigorously
presents in the first chapter sets, relations, and the algebraic-topologic
structure for R and R , properties of real functions of real argument in the
second chapter, and the concept of vector in R3, described in the third
chapter. The book presents itself as an original and synthetic approach
that allows a quick learning of the fundamental trigonometric functions and
the corresponding inversive restrictions thoroughly analyzed in the fourth
chapter, followed by the suggestive applications in the last chapter,
without any further consultation of some additional bibliography. The
theoretical considerations and the applicative issues are significant
proving to be very useful for everyday classes, different exams and the
methodic-scientific improvement. Thus, the book addresses anyone who is
interested not only in properly knowing trigonometry, but also in the major
elements regarding one of the essence of mathematics. It is the result of
the unrests of a teenager fascinated by mathematics and of the
responsibilities of the teacher he has come to be and it is also of interest
to: advanced high school students, undergraduates, and mathematics
teachers. The author shows great responsiveness towards any suggestion
meant to improve the contents and the presentation, with thanks everyone
who has made possible the publishing of this book.

**
CONFLICT OF INTEREST**

None Declare

**
ACKNOWLEDGEMENT**

None Declare

**
Vasile Postolică,**

Romanian Academy of Scientists,

“Vasile Alecsandri” University of Bacău,

România,

Email: vpostolica@ambra.ro

#### List of Contributors

##### Author(s):

**Vasile Postolică **

University of Cambridge

Kings College

Cambridge

UK