Question

Use the formula Tn(x) = cos(n cos⁻¹ x) to find the Chebyshev polynomials T0(x), T1(x), T2(x), T3(x), and T4(x).

Answer 1

The Chebyshev polynomials T0(x), T1(x), T2(x), T3(x), and T4(x) can be found using the formula Tn(x) = cos(n cos⁻¹ x). T0(x) = 1, T1(x) = cos(x), T2(x) = cos(2x), T3(x) = cos(3x), and T4(x) = cos(4x).

Step-by-step explanation:

The Chebyshev polynomials can be found using the formula Tn(x) = cos(n cos⁻¹ x).

1. For T0(x), substitute n=0 into the formula to get T0(x) = cos(0 cos⁻¹ x) = cos(0) = 1.
2. For T1(x), substitute n=1 into the formula to get T1(x) = cos(1 cos⁻¹ x) = cos(x).
3. For T2(x), substitute n=2 into the formula to get T2(x) = cos(2 cos⁻¹ x) = cos(2x).
4. For T3(x), substitute n=3 into the formula to get T3(x) = cos(3 cos⁻¹ x) = cos(3x).
5. For T4(x), substitute n=4 into the formula to get T4(x) = cos(4 cos⁻¹ x) = cos(4x).