Question 312708
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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, *[tex \LARGE \cos(3\theta)\ =\ 4cos^3(\theta)\ -\ 3cos(\theta)].

Denote the rational numbers *[tex \LARGE \mathbb{Q}].

Note that a number constructible in one step from a field K is a solution of a second-order polynomial. Note also that *[tex \LARGE \frac{\pi}{3}] radians (60 degrees, written 60°) is constructible.

However, the angle of *[tex \LARGE \frac{\pi}{3}] (60 degrees) cannot be trisected.

Note *[tex \LARGE \cos\left(\frac{\pi}{3}\right)\ =\ \cos\left(60^\circ\right)\ =\ \frac{1}{2}].

If 60° could be trisected, the minimal polynomial of *[tex \LARGE \cos\left(20^\circ\right)] over *[tex \LARGE \mathbb{Q}] would be of second order.

Note the trigonometric identity *[tex \LARGE \cos(3\theta)\ =\ 4cos^3(\theta)\ -\ 3cos(\theta)].  Now let *[tex \LARGE y\ =\ \cos\left(20^\circ\right)].

By the above identity, *[tex \LARGE \cos\left(60^\circ\right)\ =\ \frac{1}{2}\ 4y^3\ -\ 3y].

So *[tex \LARGE 4y^3\ -\ 3y\ -\ \frac{1}{2}\ =\ 0].

Multiplying by two yields So *[tex \LARGE 8y^3\ -\ 6y\ -\ 1\ =\ 0], or *[tex \LARGE (2y)^3\ -\ 3(2y)\ -\ 1\ =\ 0].

Now substitute *[tex \LARGE x\ =\ 2y], so that *[tex \LARGE x^3\ -\ 3x\ -\ 1\ =\ 0].

Let *[tex \LARGE p(x)\ =\ x^3\ -\ 3x\ -\ 1].

The minimal polynomial for *[tex \LARGE x] is a factor of *[tex \LARGE p(x)].

If *[tex \LARGE p(x)] has a rational root, by the rational root theorem, it must be 1 or -1, both clearly not roots.

Therefore *[tex \LARGE p(x)] is irreducible over *[tex \LARGE \mathbb{Q}], and the minimal polynomial for *[tex \LARGE \cos\left(20^\circ\right)] is of degree 3.

So an angle of *[tex \LARGE \frac{\pi}{3}] 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
*[tex \LARGE e^{i\pi} + 1 = 0]
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