Cyclic Sequence of Angles

In summary, the sequence \theta_1, \ldots, \theta_k, \theta_1 \ldots is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form \theta_1, \ldots, \theta_k, \theta_1 \ldots.
  • #1
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[SOLVED] Cyclic Sequence of Angles

Fix an angle [itex]\theta[/itex]. Let n be a positive integer and define [itex]\theta_n = n\theta \bmod 2\pi[/itex].

The sequence [itex]\theta_1, \theta_2, \ldots[/itex] is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form [itex]\theta_1, \ldots, \theta_k, \theta_1 \ldots[/itex].

What I would like to find out is: For which angles [itex]\theta[/itex] is the sequence [itex]\{\theta_n\}[/itex] cyclic? If for some integer m > 1, [itex]\theta_1 = \theta_m \equiv \theta = m\theta[/itex], then [itex]m\theta = \theta + 2\pi x[/itex] for some non-negative integer x. Solving for [itex]\theta[/itex], I get [itex]\theta = 2\pi x / (m - 1)[/itex]. So it seems that any rational multiple of [itex]\pi[/itex] will create a cyclic sequence. Is this correct?
 
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  • #2
Yes. Any rational multiple of pi creates a cyclic sequence. Why are you insecure about this?
 
  • #3
This all began when I started contemplating about the limit as n approaches infinity of zn, z being complex with |z| < 1. If I represent z as r(cos t + isin t), zn = rn(cos nt + isin nt). If {nt} is cyclic, then I could break up the sine and cosine terms, multiply through by rn and apply the limit on each term. Each term goes to 0 because r < 1 so the limit is 0. Right?
 
  • #4
Yes, but you don't have to worry about 'cyclic' |cos nt+i*sin nt| is bounded, since |cos|<=1 and |sin|<=1. Regardless of the arguments. So if you multiply by r^n with r<1, the result certainly goes to 0.
 
  • #5
That makes sense. So for any complex z with |z| < 1, zn goes to 0 as n goes to infinity. I began feeling paranoid about this when I was trying to compute the limit of nzn as n goes to infinity. I rewrote this as n/z-n and applied l'Hopital's rule to get -zn/log z. I wasn't sure about the limit of zn here, but now I am. Thanks.
 
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What is a cyclic sequence of angles?

A cyclic sequence of angles is a set of angles that are measured in a circular pattern, where the starting and ending points are the same. This creates a complete cycle or rotation.

What are the properties of a cyclic sequence of angles?

The properties of a cyclic sequence of angles include: all angles add up to 360 degrees, the opposite angles are supplementary, and the sum of any two adjacent angles is equal to the third angle.

What is the formula for finding the measure of an angle in a cyclic sequence?

The formula for finding the measure of an angle in a cyclic sequence is: measure of angle = (360/number of angles) * number of angles between the given angle and the starting point.

How can a cyclic sequence of angles be used in real-life applications?

Cyclic sequences of angles are commonly used in navigation, astronomy, and engineering. They can also be used in designing circular objects, such as wheels or gears.

What is the difference between a cyclic sequence of angles and a regular polygon?

A cyclic sequence of angles is a continuous set of angles that form a complete rotation, while a regular polygon is a closed shape with equal sides and angles. A cyclic sequence can have any number of angles, while a regular polygon typically has a fixed number of angles.

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