cos2theta

cos2theta denotes the cosine of twice the angle θ, written cos(2θ). It is a core double-angle function in trigonometry, arising from the sum formula for cosine and playing a central role in simplifying expressions and solving equations.

Key identities for cos(2θ) express it in terms of cosine or sine alone. The standard forms are cos(2θ) = cos^2 θ − sin^2 θ, cos(2θ) = 2 cos^2 θ − 1, and cos(2θ) = 1 − 2 sin^2 θ. Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, these expressions are equivalent. Another form is cos(2θ) = (1 − tan^2 θ)/(1 + tan^2 θ) for cos θ ≠ 0. The related identities also give cos^2 θ = (1 + cos 2θ)/2 and sin^2 θ = (1 − cos 2θ)/2, linking cos(2θ) to the basic squared-trigonometric terms.

Period, range, and symmetry: cos(2θ) has period π and takes values between −1 and 1. It is an even function of θ, since cos(2(−θ)) = cos(−2θ) = cos(2θ).

Examples: θ = 0 yields cos(0) = 1; θ = π/4 yields cos(π/2) = 0; θ = π/3 yields cos(2π/3) = −1/2.

Applications and extensions: The double-angle identity is used to simplify expressions, solve trigonometric equations, and analyze periodic signals. In algebra, cos(2θ) can be viewed as the Chebyshev polynomial of degree 2 in cos θ, namely T2(x) = 2x^2 − 1 with x = cos θ.