Intuitive understanding of the derivatives of $\\sin x$ and $\\cos x$?

Question

One of the first things ever taught in a differential calculus class: - The derivative of $\\sin x$ is $\\cos x$. - The derivative of $\\cos x$ is $-\\sin x$. This leads to a rather neat (and convenient?) chain of derivatives:

sin(x)
cos(x)
-sin(x)
-cos(x)
sin(x)
...

An analysis of the shape of their graphs confirms *some* points; for example, when $\\sin x$ is at a maximum, $\\cos x$ is zero and moving downwards; when $\\cos x$ is at a maximum, $\\sin x$ is zero and moving upwards. But these "matching points" only work for multiples of $\\pi/4$. Let us move back towards the original definition(s) of sine and cosine: At the most basic level, $\\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle. To generalize this to the domain of all real numbers, $\\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis. The definition of $\\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know. Is there anything about this **basic** definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!" That is, from **the unit circle definition alone**. Or, even more amazingly, the **right triangle definition alone**. Ignoring graphical analysis of their plot. In essence, I am asking, essentially, "Intuitively *why* is the derivative of the sine the cosine?"

Answer 1

I am asking, essentially, "Intuitively *why* is the derivative of the sine the cosine?" Because the slope of the tangent line at sin (x₀) IS cos(x₀).