Domain Of Y = Tan(x) Explained A Comprehensive Guide

The question at hand asks us to identify the domain of the trigonometric function y = tan(x). This is a fundamental concept in trigonometry and calculus, and understanding the domain of this function is crucial for various mathematical applications. To answer this question accurately, we need to recall the definition of the tangent function and its relationship to sine and cosine, and we need to identify the values of x for which the tangent function is undefined. The correct answer to the question “Which of the following describes the domain of y = tan(x), where n is any integer?” is crucial for understanding the behavior and applications of this trigonometric function. In this comprehensive guide, we will explore the intricacies of the tangent function's domain, ensuring a clear and thorough understanding of the concepts involved.

The tangent function, denoted as tan(x), is defined in terms of the sine and cosine functions. Specifically, tan(x) = sin(x) / cos(x). This definition is pivotal because it highlights that the tangent function is undefined whenever the denominator, cos(x), is equal to zero. Therefore, identifying the domain of y = tan(x) requires us to determine the values of x for which cos(x) = 0. Understanding this relationship is fundamental to grasping the behavior of the tangent function and its domain. The tangent function, as a ratio of sine to cosine, inherits characteristics from both but also possesses unique traits due to its quotient nature. This definition sets the stage for a detailed exploration of the values of x for which the function is defined and undefined.

To find the values of x where cos(x) = 0, we can refer to the unit circle or the graph of the cosine function. The cosine function represents the x-coordinate of a point on the unit circle, and it equals zero at angles where the point lies on the y-axis. These angles are π/2, 3π/2, 5π/2, and so on, as well as their negative counterparts –π/2, –3π/2, –5π/2, and so on. In general, cos(x) = 0 when x is an odd multiple of π/2. This is a crucial observation, as these are the points where the tangent function will be undefined. Identifying these values is paramount to correctly defining the domain of y = tan(x). The cosine function's zeros are spaced periodically, which is a characteristic we will leverage to express the domain in a concise mathematical form. By understanding the periodicity of cosine, we can generalize the points of discontinuity for the tangent function.

Since tan(x) is undefined when cos(x) = 0, the domain of y = tan(x) includes all real numbers except those where x is an odd multiple of π/2. Mathematically, we can express this condition as x ≠ (2n + 1)π/2, where n is any integer. This expression captures all the values where the tangent function is undefined. It’s a compact and precise way to describe the domain, accounting for the periodic nature of the cosine function and, consequently, the tangent function. Understanding this notation is key to interpreting the domain in mathematical contexts. The domain of the tangent function, therefore, consists of intervals between these points of discontinuity, highlighting its periodic yet discontinuous nature.

This can be further simplified to x ≠ π/2 + nπ, where n is any integer. This notation clearly illustrates that the domain excludes values that are π/2 plus any integer multiple of π. It encapsulates all points where the tangent function is undefined due to the cosine function being zero. The domain of a function is a fundamental aspect of its definition, and in the case of tan(x), it dictates where the function is well-behaved and where it approaches infinity or negative infinity. The ability to express this domain mathematically is crucial for advanced mathematical analysis.

Now, let's examine the given options in the context of our derived domain:

• A. x ≠ nπ: This option excludes all integer multiples of π. While these values are important for understanding the sine function (where sin(x) = 0), they are not the values that make cos(x) = 0. Therefore, this option is incorrect.
• B. x ≠ 2nπ: This option excludes even multiples of π. Again, these are not the values that make cos(x) = 0, so this option is also incorrect.
• C. x ≠ nπ/2: This option excludes all multiples of π/2, both even and odd. This is close but not entirely accurate because it includes values like π, which are in the domain of tan(x). Hence, this option is incorrect.
• D. x ≠ π/2 + nπ: This option correctly excludes all odd multiples of π/2, which are the values where cos(x) = 0. Therefore, this is the correct answer.

The domain of the function y = tan(x) is all real numbers except x = π/2 + nπ, where n is any integer. This is because the tangent function is defined as the ratio of sine to cosine, and it is undefined when the cosine function equals zero. Understanding this domain is crucial for working with trigonometric functions in various mathematical contexts. Option D, x ≠ π/2 + nπ, accurately describes the domain of y = tan(x), making it the correct answer. Mastering this concept provides a solid foundation for tackling more complex trigonometric problems and applications.

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