The cosine of a multiple of θ can be written as a polynomial in cos θ. For example,
cos 3θ = 4 cos3 θ − 3 cos θ
and
cos 4θ = 8 cos4 θ − 8 cos2 θ + 1.
But it may or may not be possible to write the sine of a multiple of θ as a polynomial in sin θ. For example,
sin 3θ = −4 sin3 θ + 3 sin θ
but
sin 4θ = − 8 sin3 θ cos θ + 4 sin θ cos θ
It turns out cos nθ can always be written as a polynomial in cos θ, but sin nθ can be written as a polynomial in sin θ if and only if n is odd. We will prove this, say more about sin nθ for even n, then be more specific about the polynomials alluded to.
Proof
We start by writing exp(inθ) two different ways:
cos nθ + i sin nθ = (cos θ + i sin θ)n
The real part of the left hand side is cos nθ and the real part of the right hand side contains powers of cos θ and even powers of sin θ. We can convert the latter to cosines by replacing sin2 θ with 1 − cos2 θ.
The imaginary part of the left hand side is sin nθ. If n is odd, the right hand side involves odd powers of sin θ and even powers of cos θ, in which case we can replace the even powers of cos θ with even powers of sin θ. But if n is even, every term in the imaginary part will involve odd powers of sin θ and odd powers of cos θ. Every odd power of cos θ can be turned into terms involving a single cos θ and an odd power of sin θ.
We’ve proven a little more than we set out to prove. When n is even, we cannot write sin nθ as a polynomial in sin θ, but we can write it as cos θ multiplied by an odd degree polynomial in sin θ. Alternatively, we could write sin nθ as sin θ multiplied by an odd degree polynomial in cos θ.
Naming polynomials
The polynomials alluded to above are not arbitrary polynomials. They are well-studied polynomials with many special properties. Yesterday’s post on Chebyshev polynomials defined Tn(x) as the nth degree polynomial for which
Tn(cos θ) = cos nθ.
That post didn’t prove that the right hand side is a polynomial, but this post did. The polynomials Tn(x) are known as Chebyshev polynomials of the first kind, or sometimes simply Chebyshev polynomials since they come up in application more often than the other kinds.
Yesterday’s post also defined Chebyshev polynomials of the second kind by
Un(cos θ) sin θ = sin (n+1)θ.
So when we say cos nθ can be written as a polynomial in cos θ, we can be more specific: that polynomial is Tn.
And when we say sin nθ can be written as sin θ times a polynomial in cos θ, we can also be more specific:
sin nθ = sin θ Un−1(cos θ).
Solving trigonometric equations
A couple years ago I wrote about systematically solving trigonometric equations. That post showed that any polynomial involving sines and cosines of multiples of θ could be reduced to a polynomial in sin θ and cos θ. The results in this post let us say more about this polynomial, that we can write it in terms of Chebyshev polynomials. This might allow us to apply some of the numerous identities these polynomials satisfy and find useful structure.