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?"