Answer to Question #140930 in Quantitative Methods for Fredrick Moleleki

Question

(a) Using the formula Tn(x) = cos(n cos−1 x), n ≥ 0, find the Chebyshev
polynomials T0(x), T1(x), T2(x), T3(x), and T4(x).

Answer 1

$T_0(x)=cos(0cos^{-1}x)=1$

$T_1(x)=cos(cos^{-1}x)=x$

$T_2(x)=cos(2cos^{-1}x)=2cos^2(cos^{-1}x)-1=2x^2-1$

$T_3(x)=cos(3cos^{-1}x)=4cos^3 (cos^{-1}x)-3 cos (cos^{-1}x)= 4x^3-3x$

$T_4(x)=cos(4cos^{-1}x)=2cos^2(2 cos^{-1}x) -1=2(2cos^2(cos^{-1}x)-1)^2-1=8cos^4 (cos^{-1}x)-8cos^2 (cos^{-1}x)+2-1= 8x^4-8x^2+1$