Now sincex2+y2= 1 we know that (sin (θ))2+ (cos (θ))2= 1. This fact is certainly worthmemorizing, but if one remembers where sin and cos come from there is no need to memorizeanything.The two periodic functions can be combined in various ways and indeed you will see this onyour assignments. However for now we will definetan(θ) =sin (θ)cos (θ).This function will also be periodic, but it is also undefined whenever cos (θ) = 0 (atπ2and-π2for example). The function gets very large in absolute value near these points.Incidentally the proper definition of a periodic function goes something like this:DEFINITION:A functionf(x) is periodic with periodLif for any integernand any validinputx,f(x+nL) =f(x).Thus we have further enriched our mathematical language with the ability to describe atleast some periodic phenomena. Of course the nicely behaved sin and cosine functions maydescribe a pendulum pretty well, but can’t possibly describe something as complex as aheartbeat.Note however that there is nothing stopping us from linking our functions together in astring of inputs and outputs, something likef(x) =p(sin (x))for a polynomialp, where we takex, feed it into the sine machine then take the output andfeed that into a polynomial function,p(.), to get another output. While it will take a coupleof years before you see why, these polynomials made up of sines and cosines let you describealmost any measurement in a compact and succinct way. Here’s a little sneak preview.Considers1(x) = cos (x),s2=s1+14cos (2x),s3(x) =s2+19cos (3x) and so on.Prettysimple pattern, right?You might want to check that the general formula can be writtenrecursively assn(x) =sn-1+1n2cosnx.(3)Of course we need to also know thats1(x) = cos (x), but once we know this, we can buildsn(x) for any integern. You can also use sigma notation to getsn(x) =k=nXk=11k2cos (kx).(4)18
This would be a big waste of time if it wasn’t for the result shown in the following picture.x321.50.51-1-0.5-1100-2-31 term 10 terms 200 terms Here I have plotteds1(x),s10(x) ands200(x) for-π < x < π. You can see thats1(x) is justcosx, but thats10ands200are essentially the same. This means that if I had a functionthat was kind of pointy nearx= 0 but round nearx=±πthen instead of finding some newform for this function I could just uses200(x) as anapproximation(actually most of thetime I could get away withs10(x)). All it takes is a recipe for choosing the numbers in frontof cos (x), cos (2x), cos (3x) and so on.You might be thinking this whole business ofp(cos (x)) is too complicated, why not just usepolynomialsp(x) to approximate? Toward this end consider the following example that youwill be able to understand in full by the end of first year. I claim that forxpretty close tozero (can you make this more precise?) that I can approximate cos(x) by the polynomialpn(x) =k=nXk=0(-1)kx2k(2k)!