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This essential trains for: AMC-12, AIME.
This is an equation in which the solutions are n-th roots of a complex number.
Given a complex number z, the equation:
In order to solve this equation we have to write 1 in trigonometric (or exponential) form because in any of these two forms we can easily take the n-th root using the de Moivre formula.
The absolute value of 1 is 1 and the argument 0. Taking into account the periodicity of the trigonometric functions:
apply de Moivre:
is 1, the one real root of the equation.
The next root:
can be represented as a point on the unit circle that is one n-th of a complete rotation in counter-clockwise direction.
Each subsequent root will be rotated by the same arc and the n-th root will finally coincide with the first one:
which means that there are only n distinct values of the argument of the generic root. Therefore, there are only n distinct roots.
These roots all have the same absolute value, which means they are all points on a unit circle in the complex plane.
The roots are placed on the circle at constant intervals of:
effectively "cutting the circle" into n congruent sectors. Hence, the name "cyclotomic."
Example 1: Find the roots of the equation:
The roots are the vertices of a regular pentagon inscribed in the unit circle.
There is one real root and 2 pairs of complex conjugate roots.
Properties of the roots
According to Viète's relations, the sum of the roots must be zero:
which is, in fact, the same as:
since the imaginary parts are pairs of opposite numbers and add up to zero.
The product of all the roots must be equal to:
The sum of the squares of the lengths of all the diagonals of the regular n-gon formed, is 2n. Proof:
The square of the length of the diagonal from a k-th root to the 0-th root, which is 1+0i is:
add over all diagonals:
and use Viète's relation for the sum of the roots:
the non-real roots of the cyclotomic equation are the roots of: