How to obtain the co-efficient of a trigonometric equations?

I have a polynomial equation like consider f(x) = x^6+5x^4+x^2, and x is 2cos(A/2). After substituting in the equation, the equation is solved in terms of a0+a1*cos(A)+a2*cos(2A)+a2*cos(3A)..... I need the values for a0,a1,a2..... Do we have any instruction to obtain co-efficient in this case? Please let me know

Thank you

Answer 1

[Walter Roberson]

This can be done using indets() and some complicated type expressions. (Maple does a better job with type expressions, in my opinion.)

Easier is probably

newf = subs(f, {cos(A),cos(2*A),cos(3*A)}, {cosA, cos2A, cos3A});

Then you can use

coeff(newf, cos3A)

to find the coefficient corresponding to cos(3*A) (for example.) Just a simple change of variables.

Answer 2

[Venkata]

How to define the polynomial equation that i asked in previous question?

[Walter Roberson]

syms x A

f = x^6 + 5*x^4 + x^2;

fnew = simplify(subs(f, x, 2*cos(A/2)));

fnew2 = subs(fnew, {cos(A),cos(2*A),cos(3*A)}, {cosA, cos2A, cos3A});

Note: simplify() might not be enough to force the trig substitution in MuPad. In Maple, one would use combine(EXPRESSION,'trig') to do that substitution, but I have not yet found the MuPad equivalent of `combine/trig`

[Venkata]

after running the code you gave me, there was error as

"??? Undefined function or variable 'cosA'."

so thought that they should be used as cos(A),cos(2*A)....

but when I did that and obtained the coefficients using 'coeffs' the answer in terms of cos(A). Please let me what to do

Thank you for your response again...

[Walter Roberson]

syms cosA cos2A cos3A

[Venkata]

Hi Walter,

This is the code i have written

syms x A

syms cosA cos2A cos3A

f = 32*(x^6)-48*(x^4)+18*(x^2)-1;

fnew = simplify(subs(f, x, 1.25*cos(A/2)));

fnew2 = subs(fnew, {cos(A),cos(2*A),cos(3*A)}, {cosA, cos2A, cos3A});

OUTPUT:

>> fnew

fnew =

(15625*cos(A)^3)/1024 + (16875*cos(A)^2)/1024 + (1275*cos(A))/1024 - 999/1024

>> fnew2

fnew2 =

(15625*cosA^3)/1024 + (16875*cosA^2)/1024 + (1275*cosA)/1024 - 999/1024

>> coeffs(fnew2,cosA)

ans =

[ -999/1024, 1275/1024, 16875/1024, 15625/1024]

>> coeffs(fnew2,cos2A)

ans =

(15625*cosA^3)/1024 + (16875*cosA^2)/1024 + (1275*cosA)/1024 - 999/1024

>>

I am unable to get the coefficients

[Walter Roberson]

Well, as I indicated above:

Note: simplify() might not be enough to force the trig substitution in MuPad. In Maple, one would use combine(EXPRESSION,'trig') to do that substitution, but I have not yet found the MuPad equivalent of `combine/trig`