The domain of [tex]$y=\csc x$[/tex] is given by [tex]$x \neq \frac{\pi}{2}+n \pi$[/tex].
A. True
B. False
Answer (2)
The domain of y = csc x is determined by the values where sin x = 0 .
sin x = 0 when x = nπ , where n is an integer.
The domain of y = csc x is therefore x = nπ .
The statement that the domain is x = 2 π + nπ is False .
Explanation
Problem Analysis We are asked to determine if the domain of y = csc x is given by x = 2 π + nπ , where n is an integer.
Definition of Cosecant Recall that csc x = s i n x 1 . Therefore, the domain of csc x is all real numbers x such that sin x = 0 .
Zeros of Sine The sine function is zero at integer multiples of π , i.e., sin x = 0 when x = nπ , where n is an integer. Thus, the domain of csc x is all real numbers except x = nπ , where n is an integer.
Comparing Domains The given domain is x = 2 π + nπ . This represents all real numbers except odd multiples of 2 π . However, the correct domain is x = nπ , which represents all real numbers except integer multiples of π .
Conclusion Since the given domain x = 2 π + nπ does not match the correct domain x = nπ , the statement is false.
Examples
Understanding the domains of trigonometric functions is crucial in fields like signal processing, where cosecant functions might model certain types of signals. Knowing where these functions are undefined helps engineers avoid singularities and ensure the stability of their systems. For example, when analyzing the resonance frequencies of an electrical circuit, the cosecant function can appear in the impedance calculation, and identifying its domain is essential for predicting the circuit's behavior.
The domain of y = csc x is given by x = nπ , where n is any integer, because csc x is undefined wherever sin x = 0 . The statement that the domain is x = 2 π + nπ is thus False .
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