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I'm looking for a trigonometry text that helps develop a lot of geometric intuition and goes deep into the subject. Also some geometry problems which actually require thinking about would be in order.

Regards.

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  • $\begingroup$ Gelfand's little book is wonderful. $\endgroup$ – symplectomorphic Feb 24 '16 at 8:46
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I would recommend Hall, Knight - Elementary Trigonometry. It's not as elementary as the title suggests and is quite rigorous for a text at that level. It would fit your requirements as it uses a lot of geometric proofs where possible, and this will motivate the geometric intuition you seek.

I'll give you an example of this: in order to show that 1 radian is a constant measure for all circles, the text first insists on establishing that the ratio of the circumference of a circle to its radius (the ratio given by $\frac{C}{r}$) is always constant. It does this by using a proof where a regular polygon of $n$-sides is inscribed into two circles of differing sizes, the proof then goes on to decompose these polygons into $n$-isosceles triangles, and through various similarity arguments shows that the ratio of the circle's circumferences to their respective radii are equivalent. It then extends this argument, to show that for sufficiently large $n$ that the ratio of the perimeter of the $n$-sided polygon (the sum of all its edges) to the radii of the circle is constant, it defines that constant to be $2\pi$.

Only after doing this does it then go on to show why the angle measure 1 radian is constant for all circles. It does this by using another similarity argument, in which it compares an arc equivalent in length to the radius to an arc made by 2 right angles (i.e. $\pi$ radians or $180^\circ $). The main thrust of the argument is that the ratio of an arc equivalent to the length of the radius (i.e. the arc that subtends an angle that measures 1 radian) to a semi-circle arc is a constant, and thus the angle measure of 1 radian holds for all circles.

If this all sounds long winded to you it's not, I'm not the greatest at exposition but the author of book is and it's a quite joy to read. The exercises are also very challenging and some questions quite unconventional. It has a whole chapter dedicated to geometric proofs of certain trigonometric identities. If you're looking for geometric proof (which will motivate geometric intuition) in the context of trigonometry, this is your book. I don't know if it'll give you a deep intuition of geometry you'll probably need to consult a geometry textbook for that, but you'll gain a solid understanding of trigonometry from it. The only downside is the last time I checked the book is quite expensive second hand, but you can find it here electronically. I highly recommend it.

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Notes from Trigonometry by Steven Butler is a good intuitive book to understand trigonometry deep from it's roots.This book is especially for novices who can't even understand how Pythagorean theorem was found.

In the Preface it says that:

My major motivation for creating these notes was to talk about topics not usually covered in trigonometry, but should be.

You could also check my answer here.

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