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Symbolic acos( cos(theta) ) does not return theta.

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While cos(acos(x)) correctly returns x, acos(cos(theta)) does not return theta.
syms theta real
assume( theta >-1 & theta < 1)
a = simplify( acos(cos(theta)) )
a = 

Risposta accettata

Walter Roberson
Walter Roberson il 31 Mar 2023
syms theta real
assume( theta >-1 & theta < 1)
a = simplify( acos(cos(theta)) )
a = 
fplot(a, [-1 1])
If the identity holds then you would expect a straight line, not two lines.
But cos(-theta) = cos(theta) so cos(-1) = cos(1) and so acos(cos(-1)) = acos(cos(1)) rather than being able to distinguish -1 and 1

Più risposte (2)

John D'Errico
John D'Errico il 31 Mar 2023
Modificato: John D'Errico il 31 Mar 2023
Is it true, that acos(cos(theta)) ALWAYS returns theta? TRY AN EXAMPLE.
acos(cos(10))
ans = 2.5664
So it is not true. It works the other way of course. at least it is mathematically true.
But the range of acos is [0,pi) for real arguments, so it cannot return a number outside that interval.
fplot(@acos,[-1,1])
And so then acos(cos(theta)) is possibly best written as
abs(pi - mod(theta - pi,2*pi))
At least, that is the best I can find for an approximation.
fplot(@(theta) acos(cos(theta)),[-20,20],'r--')
hold on
fplot(@(theta) abs(pi - mod(theta - pi,2*pi)),[-20,20],'b:')
You can see the curves overlay on top of each other.
We can try syms, but simplify won't find that solution, even if I push it pretty hard, harder than what I did below:
syms T
simplify(acos(cos(T)),'steps',10,'all',true)
ans = 
Anyway, the point is, MATLAB will not return the result you expect there, because it is not true.

Paul
Paul il 31 Mar 2023
simplify does provide an option that returns theta:
syms theta real
assume( theta >-1 & theta < 1)
a = simplify( acos(cos(theta)),'IgnoreAnalyticConstraints',true)
a = 
θ

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