Domain of the Tangent Function

Oct 13, 2025 by pynkocean.com 68 views

Introduction

When dealing with trigonometric functions, it's essential to understand the concept of their domain. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will explore the domain of the tangent function, denoted as $y = \tan x$ . We will examine the options provided and determine the correct answer.

What is the Tangent Function?

The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted as $\tan x = \frac{\sin x}{\cos x}$ . The tangent function is periodic, meaning it repeats its values at regular intervals.

Domain of the Tangent Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the tangent function, the function is defined for all real numbers except when the cosine function is equal to zero. This occurs when $x$ is an odd multiple of $\frac{\pi}{2}$ .

Analyzing the Options

Let's analyze the options provided:

A. $(-\infty, \infty)$ B. $(0, \pi)$ C. all $x \neq \frac{\pi}{2} + n\pi$ D. all $x \neq n\pi$

Option A: $(-\infty, \infty)$

Option A states that the domain of the tangent function is all real numbers. However, we know that the tangent function is undefined when the cosine function is equal to zero, which occurs when $x$ is an odd multiple of $\frac{\pi}{2}$ . Therefore, option A is incorrect.

Option B: $(0, \pi)$

Option B states that the domain of the tangent function is the interval $(0, \pi)$ . However, this is not correct because the tangent function is defined for all real numbers except when $x$ is an odd multiple of $\frac{\pi}{2}$ . Therefore, option B is incorrect.

Option C: all $x \neq \frac{\pi}{2} + n\pi$

Option C states that the domain of the tangent function is all real numbers except when $x$ is an odd multiple of $\frac{\pi}{2}$ . This is the correct definition of the domain of the tangent function. Therefore, option C is the correct answer.

Option D: all $x \neq n\pi$

Option D states that the domain of the tangent function is all real numbers except when $x$ is an even multiple of $\pi$ . However, this is not correct because the tangent function is undefined when $x$ is an odd multiple of $\frac{\pi}{2}$ . Therefore, option D is incorrect.

Conclusion

In conclusion, the domain of the tangent function is all real numbers except when $x$ is an odd multiple of $\frac{\pi}{2}$ . This is represented by option C: all $x \neq \frac{\pi}{2} + n\pi$ . Therefore, the correct answer is option C.

Final Answer

The final answer is:

Frequently Asked Questions

In this article, we will address some frequently asked questions about the domain of the tangent function.

Q: What is the domain of the tangent function?

A: The domain of the tangent function is all real numbers except when $x$ is an odd multiple of $\frac{\pi}{2}$ . This can be represented as $x \neq \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

Q: Why is the tangent function undefined at odd multiples of $\frac{\pi}{2}$ ?

A: The tangent function is undefined at odd multiples of $\frac{\pi}{2}$ because the cosine function is equal to zero at these points. When the cosine function is equal to zero, the tangent function is undefined.

Q: Can the tangent function be defined at even multiples of $\pi$ ?

A: No, the tangent function cannot be defined at even multiples of $\pi$ . While the cosine function is not equal to zero at these points, the tangent function is still undefined because the sine function is equal to zero at these points.

Q: How can I determine the domain of a tangent function with a phase shift?

A: To determine the domain of a tangent function with a phase shift, you need to find the new values of $x$ that make the cosine function equal to zero. This can be done by adding the phase shift to the original values of $x$ .

Q: Can the domain of a tangent function be restricted to a specific interval?

A: Yes, the domain of a tangent function can be restricted to a specific interval. For example, the domain of the tangent function $y = \tan (x - \frac{\pi}{4})$ can be restricted to the interval $[0, \pi]$ .

Q: How can I graph the domain of a tangent function?

A: To graph the domain of a tangent function, you need to plot the values of $x$ that make the cosine function equal to zero. This can be done by plotting the lines $x = \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

Q: Can the domain of a tangent function be used to solve equations?

A: Yes, the domain of a tangent function can be used to solve equations. For example, the equation $\tan x = 0$ can be solved by finding the values of $x$ that make the cosine function equal to zero.

Conclusion

In conclusion, the domain of the tangent function is all real numbers except when $x$ is an odd multiple of $\frac{\pi}{2}$ . This can be represented as $x \neq \frac{\pi}{2} + n\pi$ , where $n$ is an integer. We hope this article has helped you understand the domain of the tangent function and how it can be used to solve equations.

Final Answer

The final answer is:

C. all $x \neq \frac{\pi}{2} + n\pi$