Trigonometric Sums

Graph for reference

Like most of my notes, this one starts with curiosity. I saw this problem and went to Desmos to solve it. Then, I got a bit curious about trigonometric functions and what that pattern would result in $$(y=\sin(x)+\sin(2x)+\sin(3x)+\sin(4x))$$; the resulting graph looked a bit like the graph of a heart rate monitor in hospitals. I used $$\sum$$ to make it easier, giving me the function $f(x)= \displaystyle\sum_{n=1}^a \sin(n\times x)$ Since Desmos can't handle infinity, I used the next best thing: $$a=99$$. This gave me a graph that was a bit strange: it was so squished together that it looked like a solid shape instead of a continuous line (imagine the graph of the derivative...); the shape looked suspiciously like the graph of the cotangent function. After messing around a bit, I found the the two graphs that contained this strange function/shape: $y=\frac{\cot\left(\frac{x}{4}\right)}{2}$

$y=\frac{\cot\left(\frac{x}{4}+\frac{\pi}{2}\right)}{2}$ I was baffled by the connection between the two functions, sine and cotangent. However, this is only half of the story.

I went through the same process with the cosine function, giving me $g(x)= \displaystyle\sum_{n=1}^b \cos(n\times x)$ $$b=99$$ This time, the graph/shape looked suspiciously like the secant/cosecant function. Once again, after some messing and mapping, I managed to get the two defining functions of this new cosine sum: $y=\frac{\csc\left(\frac{x}{2}\right)-1}{2}$

$y=\frac{\csc\left(\frac{x}{2}\right)+1}{-2}$

My question to the brilliant community of Brilliant: How on Earth are these functions related? The seemed to pop up out of nowhere.

Note by Blan Morrison
4 days, 4 hours ago

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