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

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 = 

[Torsten]

assume( theta >=0 & theta <= pi)

Answer 1

[Walter Roberson]

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

Answer 2

[John D'Errico]

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.

Answer 3

[Paul]

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