I don't really understand what you're asking. You want to "discover" sine, but that's not calculus (at least, not to begin with).
Calculus is the study of limits, rates, integrals etc.
What are you asking for exactly?
I am searching for a book on Calculus which explains in detail and with diagrams how the equation/formula/method was derived. All books I've read in the past merely explain a systematic approach towards solving a mathematical problem, but I have not learned how it works; a simple example, what is sine and how does one go about re-discovering sine?
Does such a book exist? I don't want a book on the history of mathematics, I would like a book which teaches it to me and explains how and why it works.
I want to UNDERSTAND mathematics, not just do it!
A book on mathematics then I suppose; I saw sine as a component of calculus. Have you never questioned how sine was derived and why it works? That is the stage in life I am at and I want answers . I watched a good lecture about linear algebra and it showed me that there exists people who explain mathematics and why the methods work the way they do.
The definition of sine comes from the unit circle.
Draw a circle of radius 1 on a set of Cartesian axes, centred at the origin.
Draw a segment from the origin to the circle (i.e. a radius). This creates an angle with the positive axis in the anticlockwise direction, call it .
Draw a segment from where your radius touches the circle to the axis (perpendicular to the axis).
This length is .
Similarly, the length that goes from where sine touches the axis to the origin along the axis is . It is called cosine, because as you change your angle and the sine changes, this length changes along with it. Since it works with sine, it's the cosine.
Does that make sense so far?
But it's not exactly derived from more basic principles. It's a definition: in a right triangle that also has an angle , the sine of is defined to be the length of the side opposite to divided by the length of the hypotenuse. That is the definition of sine....how sine was derived...
One book you might try is Mathematics for the Nonmathematician, by Morris Kline. Kline (1908-1992) was an excellent expositor.