Final answer:
The function csc(θ), also known as the cosecant function, is not defined for values of θ that make the sine function equal to zero. Therefore, csc(θ) is not defined for θ values of 0, π, -π, 2π, -2π, and so on.
Step-by-step explanation:
The function csc(θ), also known as the cosecant function, is not defined for values of θ that make the sine function equal to zero. In trigonometry, the cosecant of an angle is defined as the reciprocal of the sine of that angle.
The function f(θ)=cscθ is not defined when the sine of theta, sin(θ), is equal to zero because cosecant, csc(θ), is the reciprocal of sine. Since division by zero is undefined in mathematics, any angle where sin(θ)=0 will result in the cosecant function being undefined.
Angles for which sin(θ)=0 occur at integer multiples of π (pi) radians, or, in degrees, at integer multiples of 180°. Specifically, this happens at 0, ±π, ±2π, ±3π, ..., or in degrees, at 0°, ±180°, ±360°, ±540°, and so on. Therefore, f(θ)=cscθ is not defined at θ = nπ where n is any integer. The sine of an angle is equal to zero at the x-intercepts of the sine graph, which occur at integer multiples of π.