After reading Wikipedia and some previous questions asked in this site, I still don't understand this.
Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). The conclusion was that p(x) is irreducible over integers for Angle 60 degrees( clearly trisecting that would give 20 degrees of it was solvable ).
I want to see the actual figure with explanation of how trisecting angle is equivalent to constructing a SEGMENTS such that their length ratio is cos(theta ) .
Also, is proving for only 60 degrees enough to conclude that it can't be trisected or equivalently solve p(x) ?
Note: Haven't done Galois theory and Fields. Thank you for any info.
Here is the link: https://en.m.wikipedia.org/wiki/Angle_trisection
Regarding equivalence of trisecting an angle and constructing segments in the ratio $\cos(\theta)$:
To trisect the angle $3\theta$ means that you must construct an angle equal to $\frac13(3\theta)$, that is, $\theta$.
If you can trisect $3\theta$, that is, construct angle $\theta$, you can construct a right triangle with angle $\theta$ at one vertex. The length of the leg adjacent to the angle $\theta$ is $\cos(\theta)$ times the length of the hypotenuse. So you will then have two segments (the leg and hypotenuse) whose lengths have ratio $\cos(\theta)$.
Conversely, if you can somehow construct two segments whose lengths are in the ratio $\cos(\theta)$, you can construct a right triangle using the shorter of these segments as one leg of the triangle and the longer segment as the hypotenuse. The angle between these two sides will then be $\theta$, and you will have trisected $3\theta$.
Regarding whether it's enough to show that $60$ degrees cannot be trisected: yes.
We know of course that there are certain angles we can trisect. If you happen to know that an angle is equal to $180^\circ$, or $90^\circ$, or $45^\circ$, for example, you can easily trisect it. But when people speak of the problem of "trisecting the angle with straightedge and compass," they are asking whether there is a procedure using only compass and straightedge that is guaranteed always to construct one-third of a given angle, no matter what angle you are given. "No matter what angle you are given" does not mean "no matter what angle you are given, unless it's $60$ degrees."