But unfortunately this argument is circular (I think), since the series comes from a Taylor series, which is generated using the first derivative (and the second, third and more derivatives as well).
In this post, I want to do two things. Strang has a nice picture which makes clear the relationship between position on the unit circle and speed. This is it:
The triangles for velocity and position are similar, just rotated by π / 2.
It is clear from the diagram that at any point, the y-component of the velocity is cos(t), while the y-position is sin(t). Thus, the rate of change of sin(t) is cos(t). This is the result we've been seeking. Similarly, the rate of change of the x-position, cos(t), is -sin(t).
Strang also derives this result more rigorously starting on p. 64. That derivation is a bit complicated, although not too bad, and I won't follow the whole thing here. It uses his standard approach as follows:
Applying a result found using just the Pythagorean theorem earlier (p. 31) for sin (s + t):
He comes up with this expression for:
The problem is then to determine what happens to these two expressions in the limit as h -> 0. The first one is more interesting. As h gets small, | cos(h)-1 | gets smaller like h2, so the ratio goes to 0.
R can help us see better.
Here is the plot for the second one, which converges to 1, leaving us with simply cos(x):