Some trigonometric series intuition

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 \displaystyle \sin{x} and \displaystyle \cos{x} 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

\displaystyle \cos{x} \approx 1 \\ \sin{x} \approx x

which are also the usual small-angle approximation forms of the functions. Now, we also have

\displaystyle \frac{d}{dx} \sin{x} = \cos{x} \mbox{ and } \frac{d}{dx} \cos{x} = -\sin{x}

and we find that taking the derivative of our approximated \displaystyle \sin{x} for our approximated \displaystyle \cos{x}. Now, by taking the anti-derivative of our \displaystyle \sin{x} approximation we get

\displaystyle \int{ \sin{x} dx} \approx \int{ x dx} = \frac{1}{2}x^2 + C \approx -\cos{x}\mbox { hence } \\ \cos{x} \approx 1 - \frac{1}{2}x^2

This makes a kind of sense since looking at \displaystyle \cos{x} in the graph, near zero it’s flat but then turns down almost like a parabola with negative curvature. Similar, \displaystyle \sin{x} 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

\displaystyle \int{ \cos{x} dx} \approx \int{ 1 - \frac{1}{2}x^2 dx} = x - \frac{1}{6}x^3 + C \approx \sin{x}

With \displaystyle x = 0 \Rightarrow \sin{0} = 0 we have that \displaystyle C = 0 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 \displaystyle x for \displaystyle \cos{x} to get the desired shape, and ditto with odd powers for \displaystyle \sin{x}. Thus

\displaystyle \sin{x} = x - \frac{1}{6}x^3 + \frac{1}{120}x^5 + ... \\ \cos{x} = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + ...

These are the usual series representations we learn about, and are often used as the baser definitions of the trigonometric functions.

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