It’s often nice to take a problem from several directions at once in order to strengthen the connections between ideas. What are the series representations of the and functions? Starting from an approximation of their values near zero and using their derivative values (or rather their anti-derivative rules), we’ll try to build up some series.
Near zero, by inspection we find that
which are also the usual small-angle approximation forms of the functions. Now, we also have
and we find that taking the derivative of our approximated for our approximated . Now, by taking the anti-derivative of our approximation we get
This makes a kind of sense since looking at in the graph, near zero it’s flat but then turns down almost like a parabola with negative curvature. Similar, begins to deviate from it’s approximation by becoming lesser in the positive regime and greater in the negative regime, ie. like subtracting a cubic from it. We see that
With we have that and the subtracted cubic emerges as expected. We continue to play this back-and-forth integration game, and we expect to alternatively be adding and subtracting even powers of for to get the desired shape, and ditto with odd powers for . Thus
These are the usual series representations we learn about, and are often used as the baser definitions of the trigonometric functions.