Re: When can sin(f(x)) and cos(f(x)) be expressed as polynomials in sin(x),cos(x)?
- From: quasi <quasi@xxxxxxxx>
- Date: Sun, 27 May 2007 20:14:12 -0500
On Sun, 27 May 2007 23:38:31 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
In article <r2cf531js6j77al05hac0a6us38g7ml49p@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:
Conjecture:
If f:R->R is a continuous function such that
sin(f(x))=p(sin(x),cos(x))
cos(f(x))=q(sin(x),cos(x))
where p,q are nonconstant bivariate polynomials with real
coefficients,
Then f(x)=a*x+b*Pi where a is a nonzero integer and b is rational.
f = arcsin p
q = cos f = cos arcsin p = sqrt(1 - p^2).
p^2 + q^2 = 1. Not many polynomials
satisfying that.
Hehe.
Nice.
quasi
.
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