I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trigonometry.
Are any purely theoretical uses for trigonometry? More specifically, can trigonometric concepts (or even functions) be used to prove/disprove general mathematical conjectures?
I have been told it is used a lot in calculus, but by my (extremely) limited knowledge it mainly consists of applying calculus concepts to trigonometric functions. Is this correct?
I tell my students in differential equations (a one-semester-past-calculus class) that people have been lying to them about why trig is important! Solving triangles, who cares. Trig matters for all sorts of reasons in more advanced math.
One big deal is that the sine and the cosine both satisfy $$f(x+2\pi)=f(x);$$that is, they are functions with period $2\pi$ (in radians). The amazing and hugely important thing about that is that you can use sine and cosine to “generate” any other function with period $2\pi$. That is, leaving out a lot of technical details, if $f$ is any function with period $2\pi$ then there are constants $a_n$ and $b_n$ so that $$f(x)=a_0+(a_1\cos(x)+b_1\sin(x))+(a_2\cos(2x)+b_2\sin(2x))+\dots.$$
That’s the “Fourier series” for $f$; Fourier series are awesomely useful and important in many areas of math, theoretical and applied both.
Confession: It appears people are reading this. I can’t stand it; I told a lie above. For the record, one of the “technical details” I omitted is that it’s not actually true that every $2\pi$-periodic function can be expanded in a Fourier series.
Fourier told the exact same lie. In some sense though it was true in his day, at least truer than it is now. Because the modern notion of “function” is very different from what people thought of as a “function” a few centuries ago. And it was one of the most fruitful lies in the history of mathematics.
In the same spirit, I think that although it’s not actually true, in a pre-calculus context it’s a more appropriate statement than any of the actually true assertions it approximates.