Contextualization

Introduction

Trigonometry, a branch of mathematics that studies the relationships between the sides and angles of triangles, plays a fundamental role in various areas of knowledge, such as physics, engineering, astronomy, music, and even in health-related fields. At the core of this area of study are trigonometric functions, mainly represented by sine and cosine, which are powerful mathematical tools for describing periodic phenomena.

Trigonometric functions stand out for their periodicity and ability to model cyclical phenomena. For example, we can describe the daily temperature variation, the phases of the moon, and even the pendular motion of a pendulum using sine and cosine functions. These functions, graphically represented, provide a visual image of periodic behavior and are a central concept in mathematics that you will study and apply in this project.

Graphs of trigonometric functions are, therefore, a way to visualize how these functions work. Understanding these images is crucial to better comprehend many physical and mathematical phenomena. This project will delve deeply into these functions, their graphical representations, and their real-world applications.

Contextualization

Periodic phenomena are all around us and shape our world and daily experiences. The day-night cycle, the seasons, tides, sound, and even the rhythm of our heart are all examples of periodic phenomena that we can model with trigonometric functions. Understanding these functions and how to represent them graphically allows us not only to better understand the world around us but also to do so in a concrete and visual way.

In this digitally connected world, trigonometric functions and their graphs are also at the core of many technologies we use every day. When you listen to music on your smartphone, for example, you are benefiting from graphs of trigonometric functions that help compress and decode sound. Digital image technology, TV signals, GPS, among others, all rely on graphs of trigonometric functions.

Atividade Prática

Activity Title: Periodic Phenomena and Trigonometric Functions

Project Objective

Apply the concepts of trigonometric functions and their graphical representations to model two real-life periodic phenomena chosen by the students. Through this activity, you should deeply learn the theory of trigonometric functions, gain practice in drawing the graphs of these functions, and apply them to better understand the world around you.

Detailed Project Description

Each group must choose two periodic phenomena from the real world and model each of them with a trigonometric function. For each chosen phenomenon, you should describe the phenomenon in detail, explain why it is periodic, develop a trigonometric function that corresponds to this behavior, and draw the corresponding graph.

Required Materials

• Computer with Internet access for research and project presentation.
• Graphic design software (e.g., GeoGebra, Desmos) or graph paper to manually draw trigonometric functions.
• Scientific calculator.

Detailed Step-by-Step for Activity Completion

Step 1: Form a group of 3 to 5 students.

Step 2: Choose two periodic phenomena from the real world. These can be physical (e.g., a swinging pendulum), natural (e.g., moon phases), or even human-related (e.g., sleep/wake cycle). Research details about these phenomena, including information such as cycle duration, amplitude, vertical and horizontal displacement, etc.

Step 3: For each phenomenon, develop a trigonometric function that models this behavior. Remember to consider whether the function should be sine, cosine, or a combination of both based on the phenomenon.

Step 4: Using the chosen graphic design software or graph paper, draw the trigonometric function you created. Make sure your graph is correctly scaled and labeled.

Step 5: Write a report on the project. This report should include the following structure:

1. Introduction: Provide context for your topic, explain its relevance and real-world applications, and state the objective of your project. Briefly describe the chosen phenomena.
2. Development: Explain the theory behind trigonometric functions. Describe in detail the chosen phenomena, the process of modeling trigonometric functions, the methodology used, and the drawn graphs. Discuss the results obtained, comparing your representations with the chosen trigonometric functions.
3. Conclusion: Summarize your main points and explicitly state the learnings and conclusions drawn from the project.
4. Bibliography: Cite the sources you used to work on your project.

Project Deliverables

The final project deliverable consists of the written report, which should be done as a group, and the graphs of the trigonometric functions modeled for each chosen phenomenon. The report should follow the structure mentioned above and be 5 to 10 pages long. The graphs can be inserted in the body of the report in the Development section or attached as an appendix at the end of the document. This written documentation should be strongly connected to the practical activities of the project and should explain the work done in a detailed, clear, and concise manner.