Which of the following describes the domain of $y=\tan x$, where $n$ is any integer?
A. $x \neq n \pi$
B. $x \neq 2 n \pi$
C. $x \neq \frac{n \pi}{2}$
D. $x \neq \frac{\pi}{2}+n \pi$
Answer (1)
Recognize that tan x = c o s x s i n x , which is undefined when cos x = 0 .
Identify that cos x = 0 when x = 2 π + nπ , where n is any integer.
Conclude that the domain of y = tan x excludes these values.
The domain of y = tan x is x = 2 π + nπ , where n is any integer, so x = 2 π + nπ .
Explanation
Understanding the Problem The problem asks us to find the domain of the function y = \tahn x . We know that tan x = c o s x s i n x . Therefore, tan x is undefined when cos x = 0 . We need to find the values of x for which cos x = 0 .
Finding where cosine is zero The cosine function is zero at x = 2 π , 2 3 π , 2 5 π , and so on. In general, cos x = 0 when x = 2 π + nπ , where n is any integer.
Determining the Domain Therefore, the domain of tan x is all real numbers except x = 2 π + nπ , where n is any integer.
Selecting the Correct Option Comparing this with the given options, we see that option D, x = 2 π + nπ , matches our result.
Final Answer The domain of y = tan x is all real numbers except x = 2 π + nπ , where n is any integer. Therefore, the correct answer is x = 2 π + nπ .
Examples
Understanding the domain of trigonometric functions like y = tan x is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum, the angle of displacement can be modeled using trigonometric functions. Knowing the domain ensures that the model remains valid and avoids undefined states, allowing for accurate predictions of the pendulum's behavior.
