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Question: Conversion of a term

Question:Conversion of a term

Alfred_F 335 Maple
algebra complex-analysis
1 hour ago
0

In the attached file, the trigonometric term (2, term) is transformed into a term (3, term1) consisting of radicals. Is there a Maple procedure that can be used to reverse this process? Given an algebraic term (e.g., consisting of radicals, powers, etc.), under what conditions can it be transformed into a trigonometric form (not a Fourier series) in the sense of (3) according to (2)?test.mw
 

> 

 interface(version);

`Standard Worksheet Interface, Maple 2024.2, Windows 11, October 29 2024 Build ID 1872373`

 (1)

restart

term := 2*cos(5*arcsin((1/2)*x))

2*cos(5*arcsin((1/2)*x))

 (2)
> 

term1 := expand(term)

-3*(-x^2+4)^(1/2)*x^2+(-x^2+4)^(1/2)+(-x^2+4)^(1/2)*x^4

 (3)
> 

convert(term1, trig)

-3*(-x^2+4)^(1/2)*x^2+(-x^2+4)^(1/2)+(-x^2+4)^(1/2)*x^4

 (4)
> 

simplify(term1, trig)

(-x^2+4)^(1/2)*(x^4-3*x^2+1)

 (5)

solve(term1 = sqrt(2), x)

(1/2)*(8-2*(10+2*5^(1/2))^(1/2))^(1/2), (1/2)*(8+2*(10-2*5^(1/2))^(1/2))^(1/2), (1/2)*(8+2*(10+2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8-2*(10+2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8+2*(10-2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8+2*(10+2*5^(1/2))^(1/2))^(1/2)

 (6)
> 

evalf(solve(term1 = sqrt(2), x))

.3128689302, 1.782013048, 1.975376681, -.3128689302, -1.782013048, -1.975376681

 (7)

plot(term, x = -2.5 .. 2.5)

  

plot(term1, x = -2.5 .. 2.5)

  

NULL


 

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