What is the meaning of calculating sine of a number?

Question

When we calculate sine/cos/tan etc. of a number what exactly are we doing in terms of elementary mathematical concept, please try to explain in an intuitive and theoretical manner and as much as possible explain in the most basic mathematical way.

Answer 1

If you have a unit-radius circle centered at the origin, place yourself at $(1,0)$. Now to calculate $\sin(x)$ for the given number $x$, move counter-clockwise around the circle until you have traveled a distance $x$. Wherever you land, the $y$-coordinate is $\sin(x)$. And the $x$-coordinate is $\cos(x)$. The slope of the line connecting the origin to wherever you are is $\tan(x)$.

Answer 2

It is not the whole story, anyhow you could have a "aha!" moment.

I refer to G. A. Jennings $\,$ Modern Geometry with Applications $\,$ (1994), pp. 25-26, paraphrasing the text.

To solve practical problems, initially they drew and measured scale models of triangles; then someone realized that it could be useful to have a table of triangles and their dimensions, putting in the table the dimensions of the right triangles with the hypotenuse of length one. Sines and cosines are the lengths of the legs of those right triangles. Then the dimensions of any right triangle can be obtained by similarity multiplying the table entries by a suitable factor.
It is enough to consider only right triangles because a generic triangle can be seen as the sum or the difference of two right triangles.

Summing up, trigonometric tables are a substitute for scale drawings.