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  "body": "\n## Description\n\nThe Chebyshev polynomials of the first kind are the polynomials $T_n \\in \\Z[x]$, $n \\ge 0$, given by\n$$T_0 = 1, \\qquad T_1 = x, \\qquad T_{n+1} = 2x \\, T_n - T_{n-1} \\quad (n \\ge 1).$$\nThey are characterised by $T_n(\\cos \\theta) = \\cos(n \\theta)$ for all real $\\theta$.\nThe Chebyshev polynomials of the second kind $U_n$ satisfy the same recursion with $U_0 = 1$ and $U_1 = 2x$, and $U_n(\\cos \\theta) \\sin \\theta = \\sin((n + 1) \\theta)$.\n\n## Remarks\n\nThe first values are $T_2 = 2x^2 - 1$, $T_3 = 4x^3 - 3x$ and $T_4 = 8x^4 - 8x^2 + 1$.\nCheck for $T_2$: $\\cos(2\\theta) = 2 \\cos^2 \\theta - 1$.\nFor $n \\ge 1$, $T_n$ has degree $n$ and leading coefficient $2^{n-1}$.\nIts roots are the $n$ numbers $\\cos \\frac{(2k - 1)\\pi}{2n}$, $k = 1, \\dots, n$, all in $(-1, 1)$.\nOn $[-1, 1]$ one has $\\abs{T_n(x)} \\le 1$, and the $T_n$ are orthogonal for the weight $(1 - x^2)^{-1/2}$: $\\int_{-1}^{1} T_m(x) T_n(x) (1 - x^2)^{-1/2} \\, dx = 0$ for $m \\ne n$.\nThey also satisfy $T_m(T_n(x)) = T_{mn}(x)$.\nTheir roots are good nodes for [[polynomial interpolation|SS1YMS2NPC]] on $[-1, 1]$.\n\n## Explanations\n\n### L1 human\n\nThe formula for $\\cos(n\\theta)$ in terms of $\\cos \\theta$ is a polynomial, and this polynomial is $T_n$.\nFor example, $\\cos(3\\theta) = 4\\cos^3 \\theta - 3 \\cos \\theta$, so $T_3 = 4x^3 - 3x$.\nBecause cosines stay between $-1$ and $1$, these polynomials wave between $-1$ and $1$ on the interval $[-1, 1]$.\n",
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