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Konstantine ZelatorPublished by Brainstorm Fantasia™, Inc.
252 Unsolved Problems
201 Formulas, Facts, and Results
All rights reserved© Copyright by Konstantine Zelator,
January 2005First printed edition, February 2005
Published by Brainstorm Fantasia™, Inc.,P.O. Box 4280, Pittsburgh, PA 15203, U.S.ANo part of this work may be electronically or
mechanically (or by any other means) reproducedwithout expressed (written) permission from the
publisher, except for resonably brief quotations andexcerpts for book review purposes.
Printed at Copies at Carson,the location of Copies at Carson, Inc.,
1315 East Carson St. Pittsburgh, PA 15203
ISBN#: 0-9761810-1-0
Table of Contents
Introduction
Two error corrections
Preliminaries and listing of symbols
CHAPTER 1
Arcs and Angles
CHAPTER 2
The Six Fundamental Trigonometric Functions
1. Formulas, Results, Facts
1.1 The geometric definition of the six fundamental
trigonometric functions.
1.2 The analytic definition of the six fundamental
trigonometric functions.
1.3 Symmetry
1.4 Angles that differ by 2k π+π ; k an integer.
1.5 The Domains and Ranges of the Six Fundamental
Trigonometric Functions
1.6 Periods
1.7 The Graphs of the Six Fundamental Trigonometric Functions
1.8 The Sixteen Key Points on the Trigonometric Circle
1.9 Formulas
2. Listing of the Solved Problems
3. Solutions to the Solved Problems
Pages
(i) - (viii)
(ix)
(x) – (xiii)
1 – 18
19 – 112
19 – 60
19 – 24
24 – 27
27 – 33
33 – 39
39 – 41
42 – 46
46 – 48
48 – 54
55 – 60
60 – 63
63 – 108
Table of Contents
4. Unsolved Problems
CHAPTER 3
The Summation Formulas Trigonometric functions of
α+β, α – β, nα, nα
1. Formulas, Results, Facts
1.1 Summation Formulas
1.2 Summation Formulas for the Trigonometric numbers of the arc
expression α+β+γ
1.3 Trigonometric Numbers of Multiple Angles: sin(nα), cos(nα),
tan(nα), cot(nα); n a positive integer (Mainly for n=2,3)
1.4 The trigonometric of the angle 2α in terms of tanα
1.5 Trigonometric numbers of arcs or angles of the form nα
2. Listing of the Solved Problems
3. Solutions to the Solved Problems
4. Unsolved Problems
CHAPTER 4
Transformations of Trigonometric Functions
1. Formulas, Results, Facts
1.1 Sum-to-product formulas
1.2 Product-to-sum formulas
1.3 Applications of the sum-to-product and product-to-sum formulas
1.4 Applications to the triangle: formulas-identities involving the
angles of a triangle
109 – 112
113 – 222
113-129
113 – 120
120 – 122
122 – 126
126 – 127
128 – 129
129 – 134
135 – 219
220 - 222
223 – 290
223 – 238
223 – 224
224 – 225
225 – 229
229 – 231
Table of Contents
1.5 Special Sums
2. Listing of Solved Problems
3. Solutions to the Solved Problems
4. Listing of Unsolved Problems
CHAPTER 5
Trigonometric Equations
1. Formulas, Facts, Results
1.1 The Fundamental Trigonometric Equations
1.2 Algebraic Background/ Preliminaries
1.3 The Quadratic Equations
1.4 Homogenous Equations
1.5 Symmetric Equations
1.6 The equation a(sinx+cosx)+βsinxcosx=γ
1.7 the equation asinx+βcosx=γ
2. Listing of the Solved Problem
3. Solutions to the Solved Problems
4. Listing of Unsolved Problems
CHAPTER 6:
Trigonometric Systems of Equations
Section 1: Formulas, Facts, Results
Part 1: First Category Systems
Part 2: Second Category Systems
Section 2: Listing of the Solved Problems
231 – 238
238 – 242
242 – 287
287 – 290
291 – 476
291 – 356
291 – 293
293 – 307
307 – 323
323 - 340
340 – 346
346 – 352
352 – 356
356 – 363
363 – 464
464 – 476
477 - 637
477 – 565
478 – 529
529 – 565
566 – 569
Table of Contents
Section 3: Solutions to the Solved Problems
Section 4: Listing of the Unsolved Problems
CHAPTER 7
Some Important Trigonometric Inequalities
Section 1: Formulas, Facts, Results
Subsection1: Inequalities Involving the Sine Function
Subsection 2: Inequalities Involving the Cosine Function
Subsection 3: Inequalities Involving the Tangent and Cotangent
Functions
Subsection 4: Triangle Applications
Section 2: Listing of the Solved Problems
Section 3: Solutions to the Solved Problems
Section 4: Listing of the Unsolved Problems
CHAPTER 8
Inverse Trigonometric Functions
Section 1: Formulas, Facts, Results
Subsection 1: Preliminaries, the inverse sine, the inverse cosine,
the inverse tangent and the inverse cotangent functions
Subsection 2: Illustrations: the graphs of the inverse sine, cosine,
tangent, and cotangent functions
Subsection 3: Some basic identities
569 – 631
631 – 637
638 (blank page)
639 – 675
639 – 663
640 – 652
652 – 656
657 – 661
661 – 663
663 – 664
665 – 674
674 – 675
676 (blank page)
677 - 732
677 – 707
677 - 684
684 – 686
687 – 694
Table of Contents
Subsection 4: Nine summation formulas
Subsection 5: The inverse secant and cosecant functions
Subsection 6: A few translation formulas
Section 2: Listing of the Solved Problems
Section 3: Solutions of the Solved Problems
Section 4: Listing of the Unsolved Problems
CHAPTER 9
General facts about triangles
Section 1: Formulas, Facts, Results
Subsection 1: Right Triangles
Subsection 2: The Law of Sines, The Law of Cosines, and the Law
of Perpendicular Projections
Subsection 3: Applications of the Law of Sines and the law of
Cosines
Subsection 4: Formulas for a triangle’s secondary elements in terms
of the angles A, B, Γ, and the sidelengths α, β, γ
Section 2: Listing of the Solved Problems
Section 3: Solutions to the solved Problems
Section 4: Listing of the Unsolved Problems
Appendix: Fifty Solved Exercises on Quadratic Functions
A List of Classic Works in Trigonometry
694 – 700
701 – 706
707
708 – 710
710 – 729
730 – 732
733 – 818
733 – 765
733 – 735
735 – 740
741 – 745
746 – 765
766 – 772
772 – 814
815 – 818
A – 1 to A – 49
All rights reserved© Copyright by Konstantine Zelator,
June 2001First printed edition, October 2004
Published by Brainstorm Fantasia™, Inc.,P.O. Box 4280, Pittsburgh, PA 15203, U.S.ANo part of this work may be electronically or
mechanically (or by any other means) reproducedwithout expressed (written) permission from the
publisher, except for resonably brief quotations andexcerpts for book review purposes.
Printed at Copies at Carson,the location of Copies at Carson, Inc.,
1315 East Carson St. Pittsburgh, PA 15203
ISBN#: 0-9761810-0-2
14. NME Society Talk, Mathematics Department, Rhode Island College, May 5, 2009. Title: “Integral Triangles with a 120 angle.”
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14. πME Society Talk, Mathematics Department, Rhode Island College, May 5, 2009. Title: “Integral Triangles with a 120 angle.”15. Seminar talk, Mathematics Department, University of Pittsburgh, November 17, 2010. Talk Title: “A Potpourri of Diophantine Equations.”16. Seminar talk, Mathematics Department, University of Pittsburgh, March 17, 2011. Talk Title: “Two Exponential Diophantine Equations.”17. Seminar talk, Mathematics Department, University of Pittsburgh, March 24, 2011. Talk Title: “Fermat’s Proof of the Nontrivial Insolvability of the Diophantine Equation x4+y4=z2.
18. Seminar talk, Math Department, Bloomsburg University, Talk title: “Integral Triangles with 120 angle.”19. Mathematics, Computer Science, and Statistics Seminar Series, Bloomsburg University of Pennsylvania, November 4th, 2011. Talk Title: “Integral Triangles with a 120 angle.”20. Mathematics, Computer Science, and Statistics Seminar Series, Bloomsburg University of Pennsylvania, March 27, 2012. Talk Title: “The Diophantine Equation: arctan (1/x)+arctan (1/y) = arctan(1/k).”
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