Trig functions - transcendental?

[mathematic]

Trig functions with arguments a rational multiple of π are algebriac numbers. Are ALL others trancendental (almost all are)?

 

If you know the answer, please prove it or supply reference.

[uncool]

At least for sin and cos, no. The proof I can see goes pretty deep, into the difference between algebraic integers and algebraic numbers.

 

A number is algebraic if it is the root to a polynomial; it is an algebraic integer if it is a root for a monic polynomial. It turns out that algebraic integers are a proper subring of algebraic numbers. We can see pretty easily that for any rational number p/q, 2 cos((p/q) pi) and 2 sin((p/q) pi) are algebraic integers, as (looking only at cos for now) 2 cos((p/q) pi) = e^((p/q) i pi) + e^(- (p/q) i pi), and the latter two are algebraic integers (as they both satisfy the equation x^(2q) - 1 = 0, and that polynomial is monic). Therefore, if you take an algebraic number between 0 and 1 that is not half of an algebraic integer, it is algebraic, but can't be cosine of a rational multiple of pi. A similar proof works for sin. And you can clearly get sec and csc, as they are just inverses. I don't have an obvious proof or disproof now for tan, though.

=Uncool-

Edited May 31, 2013 by uncool

[Enthalpy]

There are too few algebraic numbers!

 

No function can take any real number as an argument - of which transcendentals outnumber algebraic ones - and take different algebraic values for each transcendental argument.

 

You may refine the explanation a little bit by taking a subset over which the function is strictly increasing for instance.

[uncool]

But that isn't the question - the question is whether the functions sin(pi*x) and cos(pi*x) create a bijection between rational numbers between 0 and 1 and algebraic numbers between 0 and 1, both of which are countable.

=Uncool-

[Enthalpy]

Yes, are "all others transcendental" was the original question.

 

sin(pi*x) a bijection between rational x and algebraic sin was an assertion, not the question:

Trig functions with arguments a rational multiple of π are algebriac numbers.

 

-----

 

As for sin(pi*n/d) being algebraic:

- Because sin(pi)=sin[d*(pi/d)] is rational and is a polynom of sin(pi/d), this latter is algebraic

- sin([n*(pi/d)] is a polynom of sin(pi/d) hence is algebraic.

[uncool]

The "all others transcendental" was the following question, as I interpreted it:

 

If cos(x) is algebraic, then is x necessarily a rational multiple of pi?

 

Coupled with the fact that cos(pi*x) is decreasing on [0, 1/2] (which is what I should have said, rather than [0, 1]) and the intermediate value theorem, that question is equivalent to the bijection question. I don't see where there was a question about the image of an uncountable set being the set of algebraic numbers.

=Uncool-

[Enthalpy]

It is equivalent to the bijection question only if adding "almost all" others are transcendental, as Mathematic did properly.

[mathematic]

I posted the question on several forums. On one of them I got an answer. Rational values for sin or cos where the denominator of the fraction (in lowest terms) is not a power of two give the answer as no. Specifically the angle for such fractions is not a rational multiple of π.

 

The proof involved using a property of Chebychev polynomials of the first kind, using the trig. definition.

 

http://en.wikipedia.org/wiki/Chebyshev_polynomials

[Enthalpy]

What values do you want for sin and cos? Rational as in your last post, or algebraic like in the first?

 

By the way, cos(π/3)=0.5 is rational despite 3 not being a power of 2.

 

Could you detail what you expect from the angle and from the sine?

[mathematic]

What values do you want for sin and cos? Rational as in your last post, or algebraic like in the first?

 

By the way, cos(π/3)=0.5 is rational despite 3 not being a power of 2.

 

Could you detail what you expect from the angle and from the sine?

You misunderstood my statement. 0.5 = 1/2 has the denominator 2. That is what I was referring to, not the angle.