The cosec function which is the reciprocal of the sin function, possesses several important properties that are essential for trigonometric analysis and applications. Here are some of its key properties:
1)Periodicity: The cosec function is periodic with a period of 2π, which means it repeats its value every 2π units. This is expressed as cosec(θ+2π) = cosec(θ) for any angle θ.
2)Domain: The domain of the cosec function includes all real numbers except integer multiples of π, where cosec(θ) would be undefined due to division by zero. Thus, θ ≠ 0, ±π, ±2π,...
3)Range: The range of the cosec function is less than or equal to -1, or greater than or equal to 1. Thus, cosec(θ) ≤ -1 or cosec(θ) ≥ 1.
4)Symmetry: The cosec function is an odd function, which means that cosec(-θ) = -cosec(θ). This property indicates that the cosec function has rotational symmetry about the origin.
5)Asymptotes: The cosec function has vertical asymptotes at integer multiples of π. This means that cosec(θ) is undefined at θ = ±nπ for integers.