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Question 312708: Why you can't trisect an angle?
Answer by solver91311(24713)   (Show Source):
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The following from Wikipedia:
Angles may not in general be trisected
The geometric problem of angle trisection can be related to algebra – specifically, the roots of a cubic polynomial – since by the triple-angle formula, \ =\ 4cos^3(\theta)\ -\ 3cos(\theta)) .
Denote the rational numbers  .
Note that a number constructible in one step from a field K is a solution of a second-order polynomial. Note also that  radians (60 degrees, written 60°) is constructible.
However, the angle of (60 degrees) cannot be trisected.
Note \ =\ \cos\left(60^\circ\right)\ =\ \frac{1}{2}) .
If 60° could be trisected, the minimal polynomial of ) over would be of second order.
Note the trigonometric identity . Now let ) .
By the above identity, \ =\ \frac{1}{2}\ 4y^3\ -\ 3y) .
So  .
Multiplying by two yields So  , or ^3\ -\ 3(2y)\ -\ 1\ =\ 0) .
Now substitute  , so that  .
Let \ =\ x^3\ -\ 3x\ -\ 1) .
The minimal polynomial for  is a factor of ) .
If has a rational root, by the rational root theorem, it must be 1 or -1, both clearly not roots.
Therefore is irreducible over , and the minimal polynomial for is of degree 3.
So an angle of radians cannot be trisected.
Many people, not realizing that it is impossible, have proposed methods of trisecting the general angle, some of which provide reasonable approximations.
See http://en.wikipedia.org/wiki/Angle_trisection.
John
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