I have been through school. I have had a treatment of (Pre)Calculus (I,II,III).
I still feel like I don't know enough Trigonometry, or enough of Exponentials, or other functions.
Even through my engineering undergrad; I always felt, I lacked enough knowledge on these very essential "topics". They have been recurrent throughout my studies and had me in total surprise.
How mainstream, contemporary Calculus books enmeshes them into the real analysis topics (to me) felt like one giant over-bloated mess.
Here's the question - I want some books, which explore - Circular functions and/or Hyperbolic functions and/or Essential Exponentials. Exclusively and exhaustively.
What I've done - I have read "Introduction to Real Analysis" by Bartle and Sherbert. It was satisfying to touch actual mathematics without the advertised applications and implications. (I guess it's late transcendentals). Moreover, I'll attach this excellent book as well. I read through some trigonometry books but I am apprehensive of any geometrical methods to explain these functions now.
I am answering my own question.
I stumbled upon (and felt obliged to) attach these excellent series of blog articles, which covers the elementary functions in extensive depth and with unprecedented enthusiasm. They did succeed to satisfy me. Moreover, the author happens to be a user on MSE - @ParamanandSingh.
Throughout the series, various approaches were explored, which were all very insightful.
Exponential - 1 2 3
Circular Functions - 1 2 3
As for a book, I got interested with G.Hardy's "A Course of Pure Mathematics", especially the last two chapters. I would be appending a review soon.