Re: [Phys-L] new puzzle

[John Denker <jsd@av8n.com>]

On 5/10/22 8:49 AM, David Bowman wrote:

> [solution]

That's interesting.

I don't want to detract from the above, but I'm not sure I
agree with the following tangent:

> If the formulas are reduced to find real solutions in terms of only 
> real arithmetic then we need to be able to take trig functions of 
> fractions of arc trig functions in order to evaluate the trig 
> function of interest.

Anything you can do with complex numbers you can do with
vectors in the plane. Each vector has two real components.

In particular, if you want to find a solution for x = cos(θ),
then a method that solves for y = sin(θ) and x = cos(θ) at
the same time counts as a valid method IMHO.

Consider y to be a temporary variable introduced for convenience,
allowing geometric insight to be applied to the problem. This
sticks closer to the original question about angles.

> we are still assured that they are algebraic numbers because they can
> each be shown to obey a polynomial equation with integer
> coefficients

Agreed, yes, they are /algebraic/ by definition.

But /algebraic/ does imply nice, and doesn't imply solutions
can be found using high-school algebra (add, subtract, multiply,
divide, roots).

Abel and Galois had something to say about this.
  https://mathworld.wolfram.com/QuinticEquation.html