More juice in the lemon

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

A few days ago I first brought up the equation

\cos n\theta + i\sin n\theta = \sum_{j=0}^n {n\choose j} i^j(\cos\theta)^{n-j} (\sin\theta)^j

which holds because both sides equal exp(inθ).

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 nθ as a polynomial in cos θ,

\cos n\theta = \sum {n\choose 2k} (-1)^k (\cos\theta)^{n-2k} (1 - \cos^2\theta)^k

and in fact this polynomial is the nth Chebyshev polynomial Tn since these polynomials satisfy

\cos n\theta = T_n(\cos\theta)

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 2n + 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.

\begin{align*} \sin((2n+1)\theta) &= \sum_{\ell} (-1)^\ell {2n+1 \choose 2\ell + 1} (\cos\theta)^{2n-2\ell} (\sin\theta)^{2\ell + 1} \\ &= \sum_k (-1)^{n-k}{2n + 1 \choose 2k} (\sin \theta)^{2n + 1 -2k} (\cos\theta)^{2k} \\ &= (-1)^n \sum_k (-1)^k{2n + 1 \choose 2k} (\sin \theta)^{2n + 1 -2k} (1 -\sin^2\theta)^{k} \end{align*}

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,

 \sin((2n+1)\theta) = (-1)^n T_{2n+1}(\sin\theta)

So for all n we have

\cos n\theta = T_n(\cos\theta)

and for odd n we also have

 \sin n\theta = \pm \,\,T_n(\sin\theta)

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

More Chebyshev posts

One thought on “More juice in the lemon

  1. Peter Hancock

    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.

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