The cot function, which is the reciprocal of the tangent function, has unique properties that are important for various mathematical and practical applications. Below are some of its key properties:
1)Periodicity: The cot function is periodic with a period of π, which means it repeats its value every π units. This is expressed as cot(θ+π) = cot(θ) for any angle θ.
2)Domain: The domain of the cot function includes all real numbers except integer multiples of π, where cot(θ) would be undefined due to division by zero. Thus, θ ≠ 0, ±π, ±2π,...
3)Range: The range of the cot function is all real numbers, which means output of the cot function is between -∞ and ∞. Thus, -∞ < cot(θ) < ∞.
4)Symmetry: The cot function is an odd function, which means that cot(-θ) = -cot(θ). This property indicates that the cot function has rotational symmetry about the origin.
5)Asymptotes: The cot function has vertical asymptotes at integer multiples of π. This means that cot(θ) is undefined at θ = ±nπ for integers.