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ID: INPUT{id.yaml}

Title: >
  $\cos(\pi x)$ for rational $x$

Definition: >
  The list contains the values of $\cos(\pi x)$
  at certain rational numbers $0 \leq x \leq 1$.

Parameters:
  x:
    type: Q
    constraints: $0 \leq x \leq 1$
   
Comments:
  comment-real-parts-of-roots-of-unity: >
    For $x = 2k/n$, $\cos(\pi x)$ equals 
    the real part of the 
    $n$'th HREF{Roots_or_unity}[root of unity] $\exp(2\pi i k/n)$ CITE{WikiRootsOf1}.
  comment-roots-of-Tn: >
    For $x = (k-1/2)/n$ ($k = 1, \ldots, n$),
    $\cos(\pi x)$ equals
    the $k$'th root of the $n$'th Chebyshev polynomial of the first kind $T_n$ CITE{WikiChebyshevPoly}. 
  comment-roots-of-Un: >
    For $x = k/(n+1)$ ($k = 1, \ldots, n$),
    $\cos(\pi x)$ equals
    the $k$'th root of the $n$'th Chebyshev polynomial of the second kind $U_n$ CITE{WikiChebyshevPoly}. 
  comment-extrema-of-Tn: >
    For $x = k/n$ ($k = 0, \ldots, n$),
    $\cos(\pi x)$ equals
    the $k$'th extremum of $T_n$ CITE{WikiChebyshevPoly}. 
    
Formulas:

Programs:

References:

Links:
  WikiChebyshevPoly:
    title: "Wikipedia: Chebyshev polynomials"
    url: https://en.wikipedia.org/wiki/Chebyshev_polynomials
  WikiRootsOf1:
    title: "Wikipedia: Root of unity"
    url: https://en.wikipedia.org/wiki/Root_of_unity
    
Similar tables:
  
Keywords:
  
Tags:
- special values
- algebraic

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $\cos(\pi x)$

Numbers: INPUT{numbers.yaml}

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$\cos(\pi x)$ for rational $x$

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Numbers

$x$

$\cos(\pi x)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains the values of $\cos(\pi x)$ at certain rational numbers $0 \leq x \leq 1$.

Parameters

$x$

— rational number ($0 \leq x \leq 1$)

Comments

(1)

For $x = 2k/n$, $\cos(\pi x)$ equals the real part of the $n$'th root of unity $\exp(2\pi i k/n)$ [2].

(2)

For $x = (k-1/2)/n$ ($k = 1, \ldots, n$), $\cos(\pi x)$ equals the $k$'th root of the $n$'th Chebyshev polynomial of the first kind $T_n$ [1].

(3)

For $x = k/(n+1)$ ($k = 1, \ldots, n$), $\cos(\pi x)$ equals the $k$'th root of the $n$'th Chebyshev polynomial of the second kind $U_n$ [1].

(4)

For $x = k/n$ ($k = 0, \ldots, n$), $\cos(\pi x)$ equals the $k$'th extremum of $T_n$ [1].

Links

[1]

Wikipedia: Chebyshev polynomials

[2]

Wikipedia: Root of unity

Data properties

Entries are of type: real number

Table is complete: no