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Simple Harmonic Motion

The above mechanical example serves to introduce the idea of   $\cos(\theta)$  and   $\sin(\theta)$  as functions in the sense to which we have (I hope) now become accustomed. In particular, if we realize that (by definition)   $v_x \equiv \dot{x}$  and   $a_x \equiv \ddot{x}$,  the formulae for  v_x(t)  and  a_x(t) represent the derivatives of  x(t):

$$r \, \cos(\omega t)$$ x = (13.4)

$$\dot{x} - \, r \, \omega \, \sin(\omega t)$$ = (13.5)

$$\ddot{x} - \, r \, \omega^2 \, \cos(\omega t)$$ = (13.6)

-- which in turn tell us the derivatives of the sine and cosine functions:

$${d \over dt} \, \cos(\omega t) - \omega \, \sin(\omega t)$$ = (13.7)

$${d \over dt} \, \sin(\omega t) \omega \, \cos(\omega t)$$ = (13.8)

So if we want we can calculate the   $n^{\rm th}$  derivative of a sine or cosine function almost as easily as we did for our ``old'' friend the exponential function. I will not go through the details this time, but this feature again allows us to express these functions as series expansions:

$$\begin{array}[c]{rcccccc} \exp(z) =& 1 &+ z &+ \onehalf z^2 . . . . . . \sin(z) =& & z & &- {1 \over 3!} z^3 & &+ \cdots \end{array}$$ (13.9)

where I have shown the exponential function along with the sine and cosine for reasons that will soon be apparent.