Multiple angle asymmetry

The cosine of a multiple of can be written as a polynomial in cos . For example,

cos 3 = 4 cos³ - 3 cos

and

cos 4 = 8 cos⁴ - 8 cos² + 1.

But it may or may not be possible to write the sine of a multiple of as a polynomial in sin . For example,

sin 3 = -4 sin³ + 3 sin

but

sin 4 = - 8 sin³ cos + 4 sin cos

It turns out cos n can always be written as a polynomial in cos , but sinn can be written as a polynomial in sin if and only if n is odd. We will prove this, say more about sin n for evenn, then be more specific about the polynomials alluded to.

Proof

We start by writing exp(in) two different ways:

cos n + i sin n = (cos + i sin )ⁿ

The real part of the left hand side is cos n and the real part of the right hand side contains powers of cos and even powers of sin . We can convert the latter to cosines by replacing sin² with 1 - cos² .

The imaginary part of the left hand side is sin n. If n is odd, the right hand side involves odd powers of sin and even powers of cos , in which case we can replace the even powers of cos with even powers of sin . But ifn is even, every term in the imaginary part will involve odd powers of sin and odd powers of cos . Every odd power of cos can be turned into terms involving a single cos and an odd power of sin .

We've proven a little more than we set out to prove. When n is even, we cannot write sin n as a polynomial in sin , but we can write it as cos multiplied by an odd degree polynomial in sin . Alternatively, we could write sin n as sin multiplied by an odd degree polynomial in cos .

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 T_n(x) as the nth degree polynomial for which

T_n(cos ) = cos n.

That post didn't prove that the right hand side is a polynomial, but this post did. The polynomials T_n(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

U_n(cos ) sin = sin (n+1).

So when we say cos n can be written as a polynomial in cos , we can be more specific: that polynomial is T_n.

And when we say sin n can be written as sin times a polynomial in cos , we can also be more specific:

sin n = sin U_n-1(cos ).

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 could be reduced to a polynomial in sin and cos . 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.

Related posts

• Zeros of trigonometric polynomials
• Solving quadratic trigonometric equations
• Chebyshev polynomials as distorted cosines

The post Multiple angle asymmetry first appeared on John D. Cook.