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?"