U.S.$48.00

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

U.S.$36.00

Volume 1Konstantine Zelator

Published by Brainstorm Fantasia™, Inc.

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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