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Trigonometry: Notes, Problems and Exercises

Roger Delbourgo
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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Spherical trigonometry inspires the cover art of this text, but that topic only arises in the sole appendix. While that appendix is larger than any chapter by a few pages, this book is really about planar trigonometry. That focus is essential and a conscious direction here. I informally poll students coming out of trigonometry asking, “Is trigonometry about the unit circle or triangles?” I am disappointed that so many I meet leave their first encounter with this subject assuming the unit circle is “the whole point”. The unit circle does not even make a cameo appearance in this book, and indeed, while it is a powerful tool to exhibit properties and functions, it is not required. The approach here is a welcome focus on planar triangle geometry that builds nicely to the Nine Point Circle and other topics often crowded out of introductory courses due to the time and attention paid to the unit circle’s possibilities.

Unfortunately, necessary concepts like ratio-determining similarity (a key concept for understand trigonometric definitions and proofs as well as solving many triangle problems), congruence, tangents, medians, and bisectors are used too frequently without motivation and definition. Some concepts, like collinearity, are explained as necessary for the target audience of undergraduates or high school students. These preliminary definitions, when present, only underscore the absence of others.

About two-thirds of the way through the text, a nicely interlocking series of evolving concepts gives way to bolted-on vignettes on Morley's Trisector Theorem, cyclic quadrilaterals, Ptolemy’s Theorem, etc. The flow of exposition devolves into a gallery of trigonometric curiosities. These deficiencies could all be corrected with more content, more context, and linking concepts. About 130 pages of main content are here divided into 37 brief, subject-specific chapters of only two or three pages each including exercises.

The lack of coherence and scope prevents recommending this as a textbook, but it has the content, examples, and exercises to supplement a standard text or lectures. The text would be better as a study guide or reference with a subject index. It has only an author’s index. (The table of contents works nearly as well, since the chapters are so concise and focused.) For teachers, many of the small chapters are readymade classroom capsules with one or two exercises to fill out a lecture or augment another text. While I would recommend for such usage chapters on such expected topics as Law of Cosines and the reciprocal trig functions, I think the greater value is on subjects not appearing in many introductory texts (crowded out by unit circle), such as Ptolemy’s Theorem, Napoleon Circles, and pedal triangles.

Tom Schulte teaches mathematics at Oakland Community College in Michigan and prefers the rarely seen blackboard to the prevalent whiteboard. 

  • Preface
  • Introduction: A Review of Some Geometrical Ideas
  • Pythagoras' Theorem
  • Degrees versus Radians
  • Cartesian and Polar Coordinates. The Sine and Cosine Ratios
  • The Cosine Rule
  • Stewart's Theorem. Medians and the Centroid G
  • The Circumcentre, O. The Sine Rule
  • Area. Hero's Formula
  • The Tangent Ratio
  • Some Very Special Angles
  • Cosecant, Secant, Cotangent. Proving Simple Identities
  • Further Problems — Heights & Distances
  • The Factorisation Formulae & Napier's Tangent Rule
  • Addition Formulae for Sines & Cosines
  • Addition Formulae for Tangents
  • Further Half-Angle Formulae
  • Solving the Equation a sin θ + b cos θ = c
  • Ptolemy's Theorem
  • Morley's Trisector Theorem
  • Cyclic Quadrilaterals and Brahmagupta's Formula
  • Graphs of the Six Trigonometrical Ratios
  • Graphs of the Six Inverse Trigonometrical Ratios
  • Addition Formulae for Inverse Functions & Rutherford's and Machin's Formulae
  • Solving Simultaneous Equations
  • The Problem of Elimination
  • Angle Bisectors and the Incentre, I
  • Altitudes, the Orthocentre, H and the Pedal Triangle
  • The Distances OIv, OH, IH
  • External Angle Bisectors and the Ex-Centres Ia, Ib, Ic
  • The Distances AIa, OIa, HIa
  • The Nine-Point Centre, N. The Feuerbach Circle
  • The Distances IG, IN
  • Napoleon Circles and the Fermat Point
  • Small Angles in Radians
  • Euler's Product. A Limit for π
  • Angle of Dip and Distance to Horizon
  • A Series for π
  • Stricter Inequalities. More Limits
  • Appendix A Elementary Spherical Trigonometry
  • Author Index
  • Book References
  • Appendix B More Exercises for Practice
  • Appendix C Solutions to the More Exercises for Practice