They’re all the same, but different
Weierstrass. It sounds like a pretty complicated word, but what exactly is it? Weierstrass actually refers to Karl Weierstrass, a 19th century German mathematician, who is often cited as the father of modern analysis. Incidentally, a special theorem known as the Weierstrass factorization theorem, the focus of this article, is also named after this mathematician. So, what is the Weierstrass factorization theorem?
The textbook definition of the Weierstrass factorization theorem is that every entire function can be represented as a product involving its zeroes. The definition of an entire function itself can be very complicated, but it is not of extreme relevance for this article - we just need to know that both sin(x) and cos(x) are entire functions, which means the Weierstrass factorization theorem can be applied to them. Thus, we can represent this definition mathematically as:
Using this definition, we can create a polynomial expression for any entire function. So, let us try to derive a Weierstrass expression for sin(x).
This formula gives us the Weierstrass factorized form for sin(x), which leads us to the logical question: what form could cos(x) take? Let’s look at this in 2 different ways:
This gives us one expression for a Weierstrass expansion of cos(x), which is also the more widely accepted expression for the function. However, if we try to derive an expression for this using a method similar to that used to derive the expansion for sin(x), something extremely confusing and wonderful at the same time happens. Let us look at that as well.
This is very interesting! By looking at the same problem from 2 different perspectives, we’ve received 2 different formulations for the same function! To highlight this paradox, let’s equate the two expressions:
This evidently can’t be correct! But, what did we actually do wrong? Well, to add a little more spice to the mix, let’s try to evaluate cos(x) in another way:
Looks like we’ve received yet another expression for cos(x)! But, this also gives us a small insight into a possible flaw in our method. Perhaps, we get a different formulation depending on which factor we take into account in the denominator? Let’s examine this idea using sin(x):
From here, it looks like we can get 2 different values of A for both sin(x) and cos(x)! And funnily enough, if equations (3) or (4) can be assumed to be correct, we can use sin(x) = sin(2x)/2cos(x) to get 2 more expressions for sin(x) and another expression for cos(x) as well by using equation (5)! This means that we can have a collection of 4 different expressions for both sin(x) and cos(x) as follows:
Out of these expressions, literature has shown that equations (1) and (2) are the generally accepted expressions for sin(x) and cos(x). But what exactly is wrong with the other 3 expressions for both? I’ll leave that for you guys to find out! As mathematicians love to say: “the proof is left as an exercise to the readers!”
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