The Chebyshev’s polynomials are defined by Tn(x) = cos (n arcos x) n = 0, 1, 2, 3,, .
(a) What are the domain and range of these functions?
(b) We know that T0(x) = 1 and T1 (x) = x. Express T2 explicitly as a quadratic polynomial and T3 as a cubic polynomial.
(c) Show that, for n > 1,
(d) Use part (c) to show that Tn is a polynomial of degree n.
(e) Use parts (b) and (c) to express T4, T5, T6, and T7 explicitly as polynomials.
(f) What are the zeros of Tn? At what numbers does Tn have local maximum and minimum values?
(g) Graph T2, T3, T4, and T5 on a common screen.
(h) Graph T5, T6, and T7 on a common screen.
(i) Based on your observations from parts (g) and (h), how are the zeros of Tn related to the zeros of Tn+1? What about the -coordinates of the maximum and minimum values?