How To Find The Period Of A Tangent Graph | Step-By-Step Guide

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To find the period of a tangent graph, first note that the tangent function has vertical asymptotes at x = π/2 + nπ and x = -π/2 + nπ, where n is an integer. The period is the distance between consecutive asymptotes. To find the period, set the tangent argument equal to π/2 or -π/2 and solve for x. The distance between the two solutions is the period. For example, the period of the graph y = tan(x) is π.

The Tangent Function: Understanding the Essence

In the realm of trigonometry, the tangent function reigns supreme as the trigonometric function that unravels the intricate relationship between the opposite and adjacent sides of a right triangle. It stands as a bridge, connecting the lengths of these sides and unveiling the hidden geometrical secrets that lie within.

Imagine a right triangle, a geometrical entity composed of three sides and three angles. The tangent function, denoted by tan, is defined as the ratio of the opposite side (the side opposite the angle of interest) to the adjacent side (the side adjacent to the angle of interest). This ratio captures the essence of the triangle's geometry, providing a numerical representation of the angle's magnitude.

Visualize a right triangle with sides labeled as follows:

  • Opposite side: The side opposite the angle of interest
  • Adjacent side: The side adjacent to the angle of interest
  • Hypotenuse: The side opposite the right angle

The tangent function for angle θ is calculated using the following formula:

tan(θ) = opposite side / adjacent side

This formula serves as a compass, guiding us through the uncharted waters of trigonometry, allowing us to determine the tangent of any angle with precision.

**Period in Periodic Functions: Unveiling the Rhythm of Oscillating Patterns**

In the realm of mathematics, functions that exhibit repetitive patterns are known as periodic functions. At the heart of their oscillatory nature lies a fundamental property called period, which measures the distance between consecutive peaks or troughs in their graph.

The period serves as a compass that guides us through the cyclical behavior of periodic functions. It unveils the regular intervals at which these functions reach their highest and lowest points, allowing us to predict their movements over time.

Some common examples of periodic functions include sine and cosine functions, which describe the rhythmic motion of waves or the oscillations of a spring. In these functions, the period corresponds to the wavelength or the time it takes for one complete cycle.

Understanding the period of a periodic function enables us to analyze its behavior more effectively. By identifying the period, we can anticipate when the function will repeat its pattern, allowing us to make informed predictions about its future values.

Period of the Tangent Function: Unlocking the Secrets of a Wondrous Wave

In the realm of trigonometry, where curves dance and equations sing, the tangent function stands out as a captivating entity. It's a trigonometric function that holds the key to uncovering the hidden secrets of right triangles. Its power lies in its ability to reveal the relationship between the opposite and adjacent sides of a right-angled triangle.

But what truly sets the tangent function apart is its period, a concept that unveils the rhythmic pattern inherent within its graph. The period of a function is the distance between two consecutive peaks or troughs, revealing the function's cyclical nature. In the case of the tangent function, its period is the distance between its vertical asymptotes.

Imagine the graph of the tangent function as a majestic wave, rhythmically rising and falling. The vertical asymptotes act as invisible barriers that mark the boundaries of this wave's motion. The distance between two consecutive vertical asymptotes represents the wavelength of the tangent function, the fundamental unit of its rhythmic pattern.

Understanding the period of the tangent function is crucial for unraveling the function's behavior. It allows us to predict where the function will reach its maximum and minimum values and where it will exhibit its intriguing vertical asymptotes. Armed with this knowledge, we can navigate the complexities of the tangent function with confidence and precision.

Finding the Period of a Tangent Graph

What is the Period of a Function?

In mathematics, a periodic function is a function that repeats its values at regular intervals. The period of a periodic function is the distance between consecutive peaks or troughs. It represents the length of one complete cycle of the function.

Tangent Function: A Periodic Function

The tangent function is a trigonometric function that relates the opposite and adjacent sides of a right triangle. It is also a periodic function, meaning it repeats its values at regular intervals.

Period of the Tangent Function

The period of the tangent function is π, also known as 180 degrees. This means that the tangent function repeats its values every π units along the x-axis.

Method to Find the Period of a Tangent Graph

To find the period of a tangent graph, follow these steps:

  1. Identify the equation of the tangent function. The standard equation of the tangent function is tan(x).
  2. Set the tangent argument equal to π/2 or -π/2. These values represent the vertical asymptotes of the tangent function.
  3. Solve for the variable x. This will give you the distance between consecutive vertical asymptotes, which is equal to the period.

Example: Finding the Period of a Tangent Graph

Let's find the period of the tangent function tan(2x + π/4).

  1. Identify the equation: tan(2x + π/4)
  2. Set the argument equal to π/2: tan(2x + π/4) = tan(π/2)
  3. Solve for x: 2x + π/4 = π/2
  4. Subtract π/4 from both sides: 2x = π/4 - π/4
  5. Divide by 2: x = 0

The vertical asymptotes of tan(2x + π/4) occur at x = 0 and x = π/2. The distance between these asymptotes is π/2, which is the period of the function.

Finding the Period of a Tangent Graph: An Illustrative Example

In our journey through trigonometry, we've encountered the enigmatic tangent function, a trigonometric wizard that relates the opposite and adjacent sides of a right triangle. And like all things periodic, it has a secret rhythm known as its period.

What's the Period All About?

Think of the period as the heartbeat of a periodic function. It's the regular distance between consecutive peaks or troughs, like the regular intervals at which a ball bounces. Understanding the period is crucial for predicting the function's behavior and unraveling its secrets.

The Tangent's Rhythmic Dance

So, what's the period of our beloved tangent function? It's the distance between its vertical asymptotes, those points where the function shoots off to infinity like a rocket.

Unveiling the Period: A Step-by-Step Guide

Finding the period of a tangent graph is a delightful adventure, like solving a tricky puzzle. Here's a step-by-step guide to guide you through the labyrinth:

  1. Summon the Equation of the Tangent Function: Start by identifying the equation of the tangent graph you're dealing with, like the enigmatic tan(x).

  2. Set the Tangent Argument Wisely: Now, it's time to set the tangent argument equal to the magic numbers π/2 or -π/2. These values represent the vertical asymptotes, where the function dances to infinity.

  3. Solve for the Mysterious Variable: Engage in a mathematical duel and solve for the variable x. This will reveal the distance between consecutive vertical asymptotes.

  4. Eureka Moment: The distance you've found is none other than the period of the tangent function.

Real-World Example: A Tangent Adventure

Let's embark on a real-world quest to find the period of a tangent graph. Imagine you're an intrepid explorer tasked with mapping the behavior of the function tan(2x).

  1. Step 1: Equation Retrieval: You've been bestowed with the equation tan(2x).

  2. Step 2: Vertical Asymptote Rendezvous: Set tan(2x) equal to π/2: tan(2x) = π/2.

  3. Step 3: Solving the Riddle: Unleash your algebraic prowess and solve for x: 2x = π/2 + πn, where n is any integer.

  4. Step 4: Unveiling the Period: The period, the holy grail of our quest, is the distance between consecutive vertical asymptotes. In this case, it's π/2.

The Rhythm Unveiled

Marvel at the result: the period of tan(2x) is π/2. This means that the graph of tan(2x) repeats its mesmerizing pattern every π/2 units along the x-axis. This knowledge is your compass, guiding you through the intricate world of trigonometric functions.

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