Defining The Period Of Tangent Function: A Comprehensive Guide

To find the period of tan(x), identify its vertical asymptotes which divide the number line into vertical strips. The period is the horizontal distance between adjacent asymptotes. Since the first vertical asymptote is at x=0 and the next one at x=π, the period is π. Alternatively, the period can be determined as the smallest positive value of x for which tan(x) and tan(x+period) are equal. This also yields a period of π.

Understanding Vertical Asymptotes: Key to Determining Period

• Explain that the vertical asymptotes of tan(x) divide the real number line into vertical strips.

Understanding Vertical Asymptotes: The Key to Unlocking Periodicity in tan(x)

Trigonometry is a fascinating subject that unveils the secrets of angles, triangles, and their fascinating relationships. One of the most intriguing concepts in trigonometry is the determination of the period of trigonometric functions, which represents the distance traveled by a point on a function’s graph before it repeats itself.

In this journey into the world of trigonometry, we will focus on the period of the tangent function, tan(x). Understanding the concept of vertical asymptotes is crucial in unlocking the key to determining the period of tan(x).

Vertical Asymptotes: Dividing the Number Line

Vertical asymptotes are vertical lines on a function’s graph where the function is not defined. For tan(x), the vertical asymptotes occur at x = (n + 1/2)π, where n is any integer. These asymptotes divide the real number line into vertical strips, creating a framework for understanding the function’s behavior.

Period: The Distance Between Neighbors

The period of a trigonometric function is the horizontal distance between two adjacent vertical asymptotes. For tan(x), the period is the distance between two consecutive vertical asymptotes, which is π units. This means that the graph of tan(x) repeats itself every π units along the x-axis.

One Period of tan(x): A Journey of π

To visualize the period of tan(x), imagine a point moving along its graph. Starting from any point on the graph, the point will travel π units along the x-axis before reaching a point that is identical to the starting point. This journey of π units represents one period of the tan(x) function.

Finding the Period: Two Methods

Determining the period of tan(x) can be done using two methods:

• Vertical Asymptote Method: The period is the horizontal distance between any two adjacent vertical asymptotes.
• Smallest Positive Value Method: The period is the smallest positive number for which tan(x) and tan(x + period) are equal.

By applying these methods, we can determine the period of tan(x) to be π units. Understanding this concept not only empowers us with knowledge about the tangent function but also serves as a foundation for exploring more complex trigonometric concepts.

Period as the Distance Between Vertical Asymptotes

Understanding the concept of period is crucial when delving into the intricacies of trigonometric functions. Period refers to the horizontal distance between adjacent vertical asymptotes of a trigonometric graph. In the case of the tangent function, tan(x), this distance plays a pivotal role in determining its periodic behavior.

The vertical asymptotes of tan(x) divide the real number line into infinitely many vertical strips. These vertical lines occur at values of x where the tangent function is undefined, resulting in an infinite number of asymptotes. The period of tan(x) is defined as the distance between any two of these adjacent vertical asymptotes.

To visualize this concept, consider the graph of tan(x). Notice the repeating pattern of vertical asymptotes occurring at regular intervals. The distance between any two consecutive asymptotes represents the period of the function. This distance is consistent throughout the entire graph, regardless of the location of the asymptotes on the number line.

It’s important to note that the period of a trigonometric function is unique to that function. For tan(x), the period is π, meaning that the graph repeats itself after every π units in the horizontal direction. This fundamental property allows us to predict the behavior of tan(x) over any interval of the real number line.

Unveiling the Periodicity of tan(x): A Tale of π

In the world of trigonometry, where curves dance across the coordinate plane, the tangent function stands out as a fascinating entity. Its graph, a tapestry of peaks and valleys, exhibits a mesmerizing pattern that repeats over a specific interval known as its period.

The key to unlocking this periodicity lies in understanding the vertical asymptotes of tan(x). These are the vertical lines where the function becomes undefined, creating gaps in its graph. Like invisible barriers, they divide the real number line into vertical “strips.”

Strips and Cycles: The Rhythm of tan(x)

Imagine these strips as stages in a captivating performance. Each strip represents a cycle of the tangent function, a journey from one peak to the next, a valley to a peak. The horizontal distance between adjacent asymptotes marks the duration of this cycle, the interval after which the graph repeats itself.

A Period of π: The Magic Number

Remarkably, the period of tan(x) is a constant, a universal rhythm that governs its cyclical dance. This magic number is π, the ratio of a circle’s circumference to its diameter. After an interval of π units, the graph of tan(x) starts its journey anew, as if mirroring its past performance.

Visualizing the π-Periodism

To witness this phenomenon firsthand, let’s trace the graph of tan(x). Beginning at any point, move horizontally by π units. Lo and behold, you will encounter the same height on the graph, the same peak or valley. This pattern persists, confirming the π-periodicity of tan(x).

In Essence:

• The period of tan(x) is the horizontal distance between adjacent vertical asymptotes.
• The period of tan(x) is π, a constant value.
• The graph of tan(x) repeats itself after an interval of π units.

Period as the Smallest Positive Value of x

In the realm of trigonometry, the tangent function, denoted as tan(x), exhibits a fascinating property known as its period. The period of a function refers to the horizontal distance over which the function repeats itself.

Defining the Period

The period of tan(x) can be defined as the smallest positive number p for which the following equation holds true:

tan(x) = tan(x + p)

In other words, the period of tan(x) is the smallest positive value of x that results in the same value of tan(x).

Finding the Period

To determine the period of tan(x), we can employ the following steps:

1. Identify the Vertical Asymptotes: Vertical asymptotes are vertical lines where the function is undefined. For the tangent function, the vertical asymptotes occur at x = (2n + 1)π/2, where n is an integer.

2. Measure the Distance: The horizontal distance between adjacent vertical asymptotes represents the period of tan(x).

3. Determine the Smallest Positive Value: The smallest positive value of x that satisfies the equation tan(x) = tan(x + p) represents the period.

Significance of the Period

Understanding the period of tan(x) is crucial for analyzing and graphing the function. It allows us to predict the behavior of tan(x) over different intervals and to identify key features such as the range and domain.

The period of tan(x) is a fundamental concept in trigonometry that characterizes the repetitive nature of the function. By defining the period as the smallest positive value of x for which tan(x) and tan(x + p) are equal, we gain a deeper understanding of the function’s behavior and its applications in various mathematical and scientific fields.

Determining the Period of tan(x) in Practice

Understanding the period of trigonometric functions like tan(x) is essential for comprehending their behavior and graphing them accurately. Tan’s period refers to the horizontal distance between its consecutive vertical asymptotes, where the function becomes undefined.

Vertical Asymptote Method

To find the period of tan(x) using vertical asymptotes, follow these steps:

1. Identify the vertical asymptotes of tan(x), which are located at x = (n + 1/2)π for all integers ‘n’ (i.e., π/2, 3π/2, …).

2. Calculate the horizontal distance between any two adjacent vertical asymptotes. Let’s call this distance ‘d’.

3. The period of tan(x) is equal to this distance ‘d’.

Smallest Positive Value Method

An alternative method to find the period of tan(x) is to use the smallest positive value of ‘x’ for which tan(x) equals tan(x + period). This value represents the fundamental period of the function.

To find the period using this method:

1. Set tan(x) equal to tan(x + period).

2. Simplify the equation and find the smallest positive value of ‘x’ that satisfies the equality.

3. This smallest positive value represents the period of tan(x).

Example

Let’s determine the period of tan(x) using the vertical asymptote method.

The vertical asymptotes of tan(x) are located at π/2 and 3π/2. The horizontal distance between these asymptotes is π/2 – (π/2) = π.

Therefore, the period of tan(x) is π.