The Chebyshev polynomial of degree n is characterized by the property
 .  The first six polynomials with their corresponding multiple-angle formulas are



Polynomial
Multiple-angle formula


         When n is small, the Chebyshev polynomials can be computed by using the recurrence
                                        
The formula can be obtained by using the formula for the sum of cosines
.

          Chebyshev polynomials are Lissajous curves.  Indeed, a parametrisation of the curve  for  is x = cos t, y = -sin(t - p/2), 0 < t < 2p.

Chebyshev polynomials have the property
                                         .       (2)

To see this when  
         
The proof is similar when n is odd.

            

Lissajous Figures and Chebyshev Polynomials
Julio Castiņeira Merino

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