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.

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