T4cos

T4cos is a term used in trigonometry to denote the fourth-order Chebyshev polynomial applied to the cosine of an angle. By the identity T_n(cos θ) = cos(nθ) for Chebyshev polynomials of the first kind, T4(cos θ) equals cos(4θ). Consequently, T4cos x can be read as cos(4x).

Expansion follows from the explicit form of the fourth Chebyshev polynomial: T4(y) = 8y^4 − 8y^2 + 1. With y = cos x, this gives cos(4x) = 8 cos^4 x − 8 cos^2 x + 1. This polynomial representation expresses a high-multiplicity cosine in terms of powers of cos x.

Applications of this relation include algebraic simplification, deriving multiple-angle identities, and in numerical methods that use Chebyshev polynomials for function approximation or spectral methods. The T4cos notation explicitly signals the underlying Chebyshev structure when transforming cos(4x) into a polynomial in cos x.

Notes and alternatives: The term T4cos is not universally standardized; many texts simply write cos(4x). The Chebyshev perspective is particularly useful when analyzing or approximating trig functions through polynomial bases, or when decomposing higher-frequency components into lower-degree terms.

See also: Chebyshev polynomials, multiple-angle formulas, trigonometric identities.