book recommendation on geometry and trigonometry for beginners

Question

suppose someone wants to relearn geometry and trigonometry from scratch considering he has forgot whatever he has learnt about these topics in his lifetime. Which books will be good choice for that person?

As a high school student my primary target was to ease the difficulties I am facing in solving calculus problems. After that I started with relearning trigonometry but my textbooks seems to be not sufficient for what I am looking to. Then to geometry and again the same case. So it will be really appreciated if anyone could suggest me some books for this path.

A list of problems/difficulties I am facing mainly:

1. euclidean geometry
2. stuck in the conversion of trigonometric functions. I know how to do these but intuitively it is not making sense to me. So want to learn indepth
3. inverse trigononetric functions( this part is troubling me the most)
4. general solutions of circular functions. Again I know how to solve these equations at a basic level, but my problem is regarding some theoretical things like the restrictions, some methods of solving like how are they able to pinpoint exactly the actual solution not the extraneous ones, etc.

Answer 1

Hilbert, David. Foundations Of Geometry.

Coxeter, Donald. Introduction To Geometry.

Coxeter, Donald. Geometry Re-Visited.

Hobson. Trigonometry.

Answer 2

(Note: This was an answer to the original version of the question.)

I think there are many different situations where a person might be looking for a treatment of geometry or trigonometry that starts from scratch in some sense. So my answer will address various scenarios of this kind.

With a couple of exceptions, I will only mention books in English. In each case, the date is for the first edition in the original language.

1. You've never seen these subjects and you have little experience of mathematics. You are aiming, ultimately, for somewhat high problem-solving ability.

Read the following books by Durell (all of which can be downloaded from knowledge-dojo.com):

• Simplified Geometry (1931) or A New Geometry for Schools, Stage A (1939), followed by
• A New Geometry for Schools, Stage B (1939) and Elementary Trigonometry (1936).

These books were in use in England primarily from the 1930s to the 1950s. The first stage was roughly for 11- to 13-year-olds and incorporated intuitive arguments. The second stage, for 13- to 16-year-olds, provided a systematic development of geometry.

Durell later wrote a book, School Geometry and Trigonometry (1949), developing Stage B geometry and trigonometry in a unified way, but I haven't seen it.

2. You've never seen these subjects. You are aiming for modest problem-solving ability, comparable to the level of an able student who hasn't studied beyond the ordinary high school curriculum in the U.S.

• Geometry: A High School Course (1983), by Lang and Murrow, or Basic Geometry (1940), by Birkhoff and Beatley.
• Trigonometry (1999), by Gelfand and Saul, or Algebra and Trigonometry (2010), by Axler.

The book by Lang and Murrow discusses axiomatics only to a modest extent, in view of the class of readers addressed. Basic Geometry adopts an extremely strong axiom system from the outset. In both cases, proofs play an important role, but the reduced emphasis on foundations leads to discussion of substantive geometry much faster than is typical in school books, something which retains the novice reader's interest.

3. You've seen some geometry/trigonometry before, but you'd like to start over with a more systematic development of the subject. You are aiming for a high level of probem-solving ability.

• A School Geometry (1930), by Forder.
• Trigonometry, Part 1: Intermediate Trigonometry (1937), by MacRobert and Arthur.

Forder's geometry was for 13- to 16-year-olds in England. It was well-known for its challenging problems.

The first volume of Trigonometry was "intended for use in the first-year classes in the Universities and in the more advanced classes in the schools" in Scotland. Though no prior familiarity with trigonometry is strictly necessary, it seems preferable to have seen it at least in a numerical form up to the solution of triangles. Failing that, the exposition might appear to lack motivation, as the book treats analytic aspects at length before discussing geometric applications to triangles and quadrilaterals.

4. You're already proficient in geometry but would like to read an in-depth treatment of school geometry from scratch. You are aiming for very high problem-solving ability.

• Lessons in Geometry (1898), by Hadamard.

This book has been translated into many languages, but only Volume 1 on plane geometry has an English version. The final French edition appeared in 1949.

Though written as a textbook for 15- to 18-year-olds in France, in practice it was addressed only to the very best of them. The mathematician Laurent Schwartz mentions in his autobiography that he was influenced by this book in his school days.

5. You have at least a minimal acquaintance with geometry and would like to learn geometry ab ovo through an approach based on transformations, using a rigorous axiom system.

• Géométrie plane (1960), by Delessert.

This book was used with able 13- to 16-year-olds in the Swiss canton of Vaud. One of its interesting features is a character, Zosime, who engages in repeated dialogues with the author around the concepts of axioms and proof.

As in the next book by Pogorelov, the axiom system takes the idea of number for granted, is based on the concept of distance, and incorporates an equivalent of Pasch's axiom on the separation of the plane by any line into two half-planes (which is a step up from Euclid in rigor).

6. You know school geometry fairly well but would like to see how it can be developed ab ovo from a rigorous axiom system intended for use with 12-year-olds.

• Geometría elemental (1969) by Pogorelov.

This book, originally written in Russian, appears at first glance to be a textbook addressed to pupils, but it is in fact primarily addressed to schoolteachers. The book is not suitable for beginners because of the insufficient number of exercises, particularly easy ones, as well as the excessive theoretical difficulty of some sections. For example, the concept of the area of a polygon, rather than being postulated, is given a proper definition and studied rigorously.

7. You know at least a modest amount of linear algebra and would like to see elementary geometry developed mainly along analytic lines.

• Geometry: A Comprehensive Course (1970), by Pedoe.

The focus is on interesting geometry and not on foundations. Provides a substantial introduction to projective geometry.

8. You want to learn elementary geometry the way Jean Dieudonné (of Bourbaki) thought it should be taught to 15- to 18-year-olds, starting from the axioms of a vector space.

• Linear Algebra and Geometry (1964), by Dieudonné.

You will see a lot of linear algebra but little in the way of triangles, quadrilaterals, circles or conics.

9. Your mathematical sophistication is at university level, and you are interested in seeing geometry developed from (an improved version of) Hilbert's axioms in a concise, digestible form. You also want an introduction to non-Euclidean geometry.

• Lectures on the Foundations of Geometry (1959), by Pogorelov.

Includes chapters on hyperbolic and projective geometry seen from an axiomatic standpoint. In some ways, this could be compared to a slimmed down version of Efimov's Higher Geometry. (My description here is based on the fourth Russian edition, of 1979. The English translation, which I haven't seen, was from the first Russian edition.)

10. Your mathematical sophistication is at university level. You are looking for in-depth discussion of the foundations of geometry.

• Geometry: Euclid and Beyond (2000), by Hartshorne.

The book asks you to read some of Euclid's Elements while it provides commentary and a critical perspective from a modern standpoint. This is followed by extensive discussion of the foundations of geometry, on the basis of various modifications of the axiom system from Hilbert's Foundations. In comparison with Pogorelov's Lectures, there is a heavy emphasis on the consequences of weakening the completeness axioms, and there is some discussion of recent results.