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For example:

$\cos(\frac{\pi}{3}) = \frac{1}{2}$

$\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

Is there any other constant $\theta$ such that $\cos(k\theta)$ is rational or a known irrational where $k$ is not $0$ or something trivial like $\frac{\pi}{\theta}$?

transcendental
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  • what about $\theta=1$? Is this relevant? http://en.wikipedia.org/wiki/Lindemann-Weierstrass_theorem What counts as known? – snulty Apr 22 '14 at 21:31
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    How about $\theta=\arctan(2)$ and $k=1$? – BRT Apr 22 '14 at 21:35
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    Please clarify what you mean by known constant. Are imaginary numbers allowed? Cosine takes all values on $[-1,1]$; so any number, rational, irrational, or transcendental, that is on that interval, has a corresponding angle that would satisfy your question. Are values produced from inverse trigonometric functions allowed? For example, $\theta=\arctan(n)$ gives $\cos(\theta)=\frac{1}{\sqrt{n^2+1}}$, which is rational or irrational for integer values of $n$. Is that a known constant? – rajb245 Apr 22 '14 at 21:39
  • @rajb245 Sorry, I meant any number that is known to be irrational. I have not worked with the imaginary unit in trigonometric functions. The example you gave is interesting. I did not know that before. – transcendental Apr 22 '14 at 21:47
  • Here is a constant number: $\arctan(10)$. It's cosine is $\frac{1}{\sqrt{101}}$, which is a known irrational number. My question is, does the number $\arctan(10)$ qualify as a "constant", by your definition? – rajb245 Apr 22 '14 at 22:22
  • What's more, $\cos(2\arctan(10))=\frac{-99}{101}$ and $\cos(\arctan(10)/2)$ has a closed forms in terms of radicals. There are tons of forms like this if the outputs of inverse trig functions are considered constants. – rajb245 Apr 22 '14 at 22:29
  • @rajb245 Yes those kinds of constants was what I was looking for. – transcendental Apr 23 '14 at 01:01

1 Answers1

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Here is a reasonable reference: http://en.wikipedia.org/wiki/Exact_trigonometric_constants

To quote: "According to Niven's theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2, and 1."

Doing a search for "rational values of cosine" produced this result and many other useful references.

marty cohen
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