Chebyshev Polynomials - Application to Polynomial Interpolation

Recall that the Chebyshev polynomials are defined by

\[ T_n (x) = \cos ( n \arccos x ),\quad T_n ( \cos \theta) = \cos n \theta . \]

The initial values of \( T_{n} (x) \) are

\[ \begin{align} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^2 - 1 \\ T_3(x) &= 4x^3 - 3x \\ T_4(x) &= 8x^4 - 8x^2 + 1 \\ T_5(x) &= 16x^5 - 20x^3 + 5x \\ T_6(x) &= 32x^6 - 48x^4 + 18x^2 - 1 \\ T_7(x) &= 64x^7 - 112x^5 + 56x^3 - 7x \\ T_8(x) &= 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \\ T_9(x) &= 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \\ T_{10}(x) &= 512x^{10} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1. \\ \end{align} \]

For a given value \(y\) between -1 and 1, the solutions to \( T_n (x) = y \) are \( \cos \frac{ \theta + 2 \pi k } { n }\), where \( k \) ranges from 1 to \(n\) and \( \cos \theta = y \).

For each of these values, we have

\[ T_n \left( \cos \frac{ \theta + 2 \pi k } { n }\right) = \cos\left( n \times \frac{ \theta + 2 \pi k } { n } \right ) = \cos \theta = y. \]

Since we have \(n\) solutions to a degree \(n\) polynomial, these are all of the roots. \(_\square\)

In terms of \( \cos A \), what are the roots of \( T_3 ( x) = 0?\)

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In terms of \( \cos A \), what are the roots of \( 2 T_4 ( x) - 1 = 0?\)

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In terms of \( \cos A \), what are the roots of \( 4 T_5 ^2 ( x) - 3 = 0?\)

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The converse of the above theorem is as follows:

The polynomial whose roots are \( \cos \frac{ \theta + 2\pi k } { n }\), where \( k \) ranges from 1 to \(n\), is

\[ T_n ( x) = \cos \theta . \]

In terms of \( T_n \), find a polynomial whose roots are \( \cos 10 ^ \circ , \cos 100^\circ, \cos 190^ \circ, \cos 180 ^ \circ \).

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In terms of \( T_n \), find a polynomial whose roots are \( \cos \left( \theta + \frac{ 2 n } { 5} \pi \right) \) for \( i = 1 \) to \(5.\)

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To answer problems in this section, you should be familiar with Vieta's formula to help you find the sum and product of roots.

Evaluate

\[ \cos 10 ^ \circ \times \cos 50 ^ \circ \times \cos 70 ^ \circ. \]

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Evaluate

\[ \frac{ 1}{ \cos 25 ^ \circ } + \frac{ 1}{ \cos 115 ^ \circ } + \frac{ 1}{ \cos 205 ^ \circ } + \frac{ 1}{ \cos 295 ^ \circ }. \]

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In terms of \( T_n \), find a polynomial whose roots are exactly

\[ \cos^2 1 ^ \circ, \cos^2 3 ^ \circ, \cos^2 5 ^ \circ, \ldots , \cos^2 89 ^ \circ .\]

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