Transforming a polynomial into a trigonometric format tia sal2

Transforming a polynomial into a trigonometric format tia sal2

Greetings All

I'm using maple 11 and I have a polynomial and would like to convert it
into
a Trigonometric format. Is this possible?

Example:
I have a polynomial
0.00154991- 4.01371 x + 1.81197 x^2 + 8.00183 x^3 - 9.3462 x^4

How can I transform this into a trigonometric format
Example:
0.00596679 Cos[6.98132 x] + 0.00358397 Cos[7.21403 x] +
2.25013 Sin[0.232711 x] - 4.51511 Sin[0.465421 x]

Note: these aren't correct answers I just wanted to include and example

tia sal2

 

This way it makes no sense: your later expression can
not be a polynom, since it does not go to +- infinity
for large values.

However you can express it in terms of certain orthogonal
polynomials, for example Chebychev polynomials written as
cos(n*arccos(x)):

0.00154991- 4.01371 *x + 1.81197 *x^2 + 8.00183 *x^3 - 9.3462 *x^4;
OrthogonalSeries:-ChangeBasis(%,ChebyshevT(n,x));
OrthogonalSeries:-ConvertToSum(%);
convert(%,cos);
eval(%);

-2.597290090+1.987662500*x-3.767115000*cos(2*arccos(x))+
2.000457500*cos(3*arccos(x))-1.168275000*cos(4*arccos(x))

The command 'expand' should give you back your polynom.

Note that for 1 < abs(x) there is some complex stuff done
in the background, which makes the difference to your above
suggestion ( arccos(x) is not longer real, which gives some
chances for cos ):

x = 1/2+1/2*cos(2*arccos(x)), now take x = 10, then
arccos(x) = ln(10+3*11^(1/2))*I ( ~ 2.993222846*I) and
cos(of that) = ... = 10 ( ~ 9.999999999 numerical).