The domain of the function y = csc(x) is all real numbers except for the values where the sine function is equal to zero.
Understanding the Domain
The cosecant function (csc(x)) is the reciprocal of the sine function (sin(x)). This means:
- csc(x) = 1/sin(x)
Since division by zero is undefined, the cosecant function is undefined whenever the sine function is zero.
Finding the Values Where Sine is Zero
The sine function is zero at multiples of pi (π):
- sin(0) = 0
- sin(π) = 0
- sin(2π) = 0
- sin(3π) = 0
- ...
This pattern continues for all integer multiples of π.
Expressing the Domain
Therefore, the domain of y = csc(x) can be expressed as:
- x ≠ nπ, where n is any integer.
This means the domain includes all real numbers except for the values of x that are multiples of π.
Visual Representation
The graph of y = csc(x) has vertical asymptotes at each of these points where the function is undefined.
