Geometry + Trigonometry Textbook

[OldBeginning]

Basically, I want a trigonometry textbook, but a lot of the texts on trigonometry use theorems in geometry as though you are already familiar with them and simply skip proofs for these theorems (such as Pythagoras theorem, some properties of triangles, properties of quadrilaterals and triangles inscribed in a circle, etc). Now, I don't want a separate text for geometry listing these theorems and their proofs. I need a trigonometry text where geometry is the introductory part (with proofs), covering only the theorems necessary and approaching these theorems in such a way that they make the sections on trigonometry easier (kind of like how Serge Lang introduced isometries before analytic geometry).

My main problem with trigonometry are the formulas dealing with relations btw. various trig. ratios (specifically, trig. ratios of sum & difference of angles). I can learn them by heart, but no matter how much I think about them, I can't understand them conceptually. Even the proofs that use the distance formula for two points on the unit circle don't seem to help with my understanding.

So, the question is, does anyone know of a trigonometric text where everything follows logically and is hard to understand (hard to understand, because it would use concepts necessary to understand the matter clearly)?

Thank You.

 

[Snicker]

Euclidean geometry is different than algebra or calculus in that it is a highly empirical science. That is, there exists many different axiomatizations of Euclidean space which are equally good so it is not necessary to encounter any axiodeductive proof in Euclidean geometry. Visual proofs should suffice.

To prove that the interior angles of a quadralateral add up to a full circle, it is only necessary to convince oneself that the interior angles of a triangle add up to a line. This can be justified by a method I was taught in 5th grade: Cut out a paper triangle and then cut each of its three corners. Juxtapose those corners and the scraps of paper form a straight line. You can prove that a quadralateral's interior angles add up to a full circle by partitioning it into two triangles and doing a bit of algebra.

There are a lot of "proofs" to the Pythagoran theorem. My favorite is the "oragami" technique explained by Vi Hart:

A great website to learn lots of geometric theorems and their proofs at cut-the-knot: http://www.cut-the-knot.org/manifesto/index.shtml

Of course, the best way to study geometry is by getting a geometry kit, a copy of Hilbert's Geometry and the Imagination, and go to work!

For trigonometry, I'd recommend Gel'fand Trigonometry. If you want a more rigorous treatment of the trigonometric functions (i.e. their analytic definition), Shilov has a nice section on it in Elementary Real and Complex Analysis.

 

[OldBeginning]

thanks