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

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...