Cotbeta

Cotbeta, or the cotangent of beta, is a trigonometric function defined as the ratio of the cosine to the sine of an angle beta: cot(beta) = cos(beta)/sin(beta). This expression is valid for real beta whenever sin(beta) ≠ 0, i.e., beta ≠ kπ for any integer k.

Key properties include that cot(beta) is periodic with period π, so cot(beta + π) = cot(beta). Its graph consists of repeating branches between vertical asymptotes at beta = kπ. The function has zeros at beta = π/2 + kπ (where cos(beta) = 0) and vertical asymptotes at beta = kπ (where sin(beta) = 0). The cotangent is the reciprocal of the tangent function: cot(beta) = 1/tan(beta) where tan(beta) is defined.

Useful identities include cot(α + β) = (cot α cot β − 1) / (cot α + cot β) and cot(α − β) = (cot α cot β + 1) / (cot β − cot α). These relations follow from tangent or sine-cosine manipulations and are helpful in simplifying trigonometric expressions.

In calculus, cotangent has the derivative d/dx [cot x] = −csc^2 x, where csc is the cosecant function. This leads to related integrals and appears in various applications, including integrals of secant and cosecant functions.

Cotbeta appears in geometry and physics as a way to express slope-related relationships and angle-dependent ratios. In the unit circle interpretation, cot(beta) equals the ratio of the adjacent side to the opposite side for the corresponding right triangle, i.e., cos(beta)/sin(beta).