Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be worth asking about. I also had an idea similar to this (https://mathoverflow.net/questions/225327/fourier-transform-of-the-critical-line-of-zeta)
but sadly, it's already been done.

$%The following sentence is horrible to read. Note to self: Rewrite the sentence later.%$

Let $a<b$ be numbers such that  $$f(a),f(b)=0,$$ and there does not exist a number $c$ such that $a<c<b$ and $$f(c)=0.$$ For clarity, I will define zeroes satisfying these properties as **adjacent zeros**. 

Certain functions exhibit symmetries about their adjacent zeros. For example, it is trivially verified that the sine function has its critical points directly between their adjacent zeros i.e., given a minimum or maximum $c$ of the sine function, we have an interval of two adjacent zeros $a,b$ 
              $%This next part of the sentence has two nested "such that"s. Fix later to improve readability.%$
such that there exists a real number $d$ such that
$$[a,b]=[c-d,c+d].$$

Translating left or right obviously doesn't affect the truth of this statement. $%not rigorous, fix later%$ 
Therefore, the cosine function also satisfies these properties. 

However, intuitively, just from looking at various Fourier transforms (where the Fourier transform doesn't have a Dirac delta function in its domain) it seems that both the real and complex part of the Fourier Transform exhibit these properties. 

Where can I find papers about this property, or, if they don't exist, is this within the reach of me proving?