ABSTRACT
We would like to compute the derivatives of these functions. What
do we know? We know that (infinitesimal) arc length along the circle is
given by
ds = r dθ, (15.1)
but we also have the (infinitesimal) Pythagorean theorem, which tells us
that
ds2 = dx2 + dy2. (15.2)
Furthermore, from x2 + y2 = r2, we obtain
x dx + y dy = 0. (15.3)
Putting this information together, we have
r2 dθ2 = dx2 + dy2 = dx2 ( 1 +
x2
y2
) = r2
dx2
y2 , (15.4)
so that
dθ2 = dx2
y2 =
dy2
x2 , (15.5)
where we have used step. By carefully using
Figure 3.2 to check signs, we can take the square root of Equation (15.5)
and rearrange terms to obtain
dy = x dθ,
dx = −y dθ. (15.6) Finally, by inserting definitions (3.2) and (3.3) and using the fact that
r = constant, we recover the familiar expressions
d sin θ = cos θ dθ,
d cos θ = − sin θ dθ. (15.7) We have thus determined the derivatives of the basic trigonometric func-
tions from little more than the geometric definition of those functions and
the Pythagorean theorem-and the ability to differentiate simple polyno-
mials.
