Summary of Trigonometric Function: Periodicity

Introduction

Relevance of the Topic

The Periodicity of Trigonometric Functions is a central and essential concept in mathematics, with a wide range of applications in various contexts: physics, engineering, statistics, signal processing, among other fields. Understanding this topic allows not only for more effective manipulation of trigonometric functions, but is also an important step towards the discovery and exploration of more complex mathematical structures.

Contextualization

The Periodicity of Trigonometric Functions is intrinsically linked to any significant study of Trigonometry. In the high school mathematics curriculum, Trigonometry is a fundamental pillar, serving as the foundation for subsequent topics in calculus and algebra. Understanding the Periodicity of Trigonometric Functions is a vital component of this study, as this knowledge allows for the manipulation of functions on various scales, whether for the study of sound waves, circular motion, or phenomena that repeat regularly over time.

Therefore, at this moment in our journey through mathematics, it is appropriate to delve into this world of repetitions and cycles, in order to understand what the nature of these functions can reveal to us. Let's explore in depth the universe of Trigonometric Functions and their Periodicity!

Theoretical Development

Components of Trigonometric Functions

• Sin(x): This is the sine function, which associates the sine value of the angle with each angle. It has a period of 2π (or 360°), which means that the sine value repeats every 2π units of measurement.
• Cos(x): This is the cosine function, which associates the cosine value of the angle with each angle. Like the sine function, the cosine function also has a period of 2π, repeating every 2π units of measurement.
• Tan(x): This is the tangent function, which associates the tangent value of the angle with each angle. The tangent function is periodic, but its period is π (or 180°), different from sine and cosine.

Trigonometric Functions and their Graphs

• The graph of a trigonometric function is a visual representation of how the function values behave in relation to a variable (in our case, the angle x).
• The graphs of the sine and cosine functions are called 'sinusoidal waves', due to their wavelike and periodic nature. These graphs repeat exactly every 2π units of measurement, demonstrating their periodicity.
• On the other hand, the graph of the tangent function is not a sinusoidal wave, but rather a series of vertical asymptotes that repeat every π units of measurement, also demonstrating its periodicity.

Aspects of Periodicity

• Period: It is the smallest positive value that, added to the function's domain, causes the function to repeat its values. For the sine and cosine functions, the period is always 2π. For the tangent function, the period is π.
• Amplitude: It is the absolute value of the maximum value that the function reaches. For the sine and cosine functions, the amplitude is always 1.
• Horizontal Shift: It is the shift of the graph on the x-axis. This causes a change in the angle values at which the function reaches its maximum or minimum values.
• Vertical Shift: It is the shift of the graph on the y-axis. This causes a change in the value at which the sinusoidal wave is centered.

Key Terms

• Periodic Function: A function that repeats regularly at equal intervals is called a periodic function. Trigonometric functions are classic examples of periodic functions.
• Sinusoidal Wave: It is a specific type of graph that occurs naturally in many physical phenomena, and is represented by the sine and cosine functions.
• Asymptotes: They are horizontal or vertical lines that the curve of a graph approaches infinitely as it moves further away from the axis.

Detailed Summary

Key Points

• Need for Periodicity: The Periodicity of Trigonometric Functions is an intrinsic characteristic that makes them unique and highly applicable in many contexts. The ability to repeat their values at equal intervals is crucial for describing cyclical physical phenomena, such as sound waves and circular motions.

• Concept of Period: The period of a function is the smallest unit of measurement in which the function repeats. In the case of the sine and cosine functions, the period is always 2π (or 360°), and for the tangent function it is π (or 180°). This implies that, within any interval of 2π (or π, in the case of tangent), the function values will repeat.

• Horizontal and Vertical Shifts: In addition to the period, trigonometric functions can undergo shifts in the Cartesian plane, known as horizontal and vertical shifts. The horizontal shift causes a change in the location on the x-axis where the function reaches its maximum and minimum values, while the vertical shift changes the value at which the sinusoidal wave is centered.

Conclusions

• Advanced Trigonometric Manipulation: Understanding the periodicity of trigonometric functions allows for more effective manipulation of these functions and the analysis of cyclical phenomena in specific intervals. This has significant implications in various disciplines, including physics, engineering, and statistics.

• Application Beyond the Curriculum: The Periodicity of Trigonometric Functions is a concept that goes beyond the high school mathematics curriculum. It is the basis for advanced studies in calculus and algebra and has practical applications in various fields of science and engineering.

Exercises

1. Exercise 1: Given the graph of a trigonometric function, identify its period, amplitude, and whether there are any horizontal or vertical shifts.

2. Exercise 2: Let f(x) = sin(3x - π/2). What is the period, amplitude, and horizontal and vertical shifts of this function?

3. Exercise 3: Suppose the motion of a particle is described by the time function given by d(t) = 5sin(2t + π/3). What is the period of this motion? And the horizontal or vertical shift in the trigonometric function?