There’s more juice left in the lemon we’ve been squeezing lately.

A few days ago I first brought up the equation

which holds because both sides equal exp(*in*θ).

Then a couple days ago I concluded a blog post by noting that by taking the real part of this equation and replacing sin²θ with 1 – cos²θ one could express cos *n*θ as a polynomial in cos θ,

and in fact this polynomial is the *n*th Chebyshev polynomial *T*_{n} since these polynomials satisfy

Now in this post I’d like to prove a relationship between Chebyshev polynomials and **sines** starting with the same raw material. The relationship between Chebyshev polynomials and cosines is well known, even a matter of definition depending on where you start, but the connection to sines is less well known.

Let’s go back to the equation at the top of the post, replace *n* with 2*n* + 1, and take the imaginary part of both sides. The odd terms of the sum contribute to the imaginary part, so we sum over 2ℓ+ 1.

Here we did a change of variables *k* = *n* – ℓ.

The final expression is the expression we began with, only evaluated at sin θ instead of cos θ. That is,

So for **all** *n* we have

and for **odd** *n* we also have

The sign is positive when *n* is congruent to 1 mod 4 and negative when *n* is congruent to 3 mod 4.

Thanks John. I really enjoy your posts.

I believe that some “effects pedals” (for guitars and such) that raise or lower the input signal by an octave work by modulating (multiplying, I think) the signal (or some component of it) by itself.

Locked down in covid, I bought myself some posh German compasses, and have been working through Euclid’s Elements. Mind-blowing. Connections between golden ratios and Fibonacci series are rattling round at the back of my mind. With Chebychev polynomials in the mix, I might have to stop just carefully dropping perpendiculars, and actually think about whats going on.