Contextualization

Introduction

In this project, we will explore trigonometric functions, a fundamental concept in mathematics that has a wide range of applications in physics, engineering, computer science, statistics, economics, and many other fields. Trigonometric functions are at the heart of trigonometry studies and, as such, are essential for understanding many phenomena in the world around us.

Trigonometric functions essentially relate the angles of a right triangle to the proportions of its sides. The most common functions are sine (sin), cosine (cos), and tangent (tan), which you may have already encountered in your previous studies.

In the field of mathematics known as analysis, trigonometric functions appear as infinite and perfect functions that can model periodic phenomena such as tides, the position of the sun and moon, economic cycles, and electromagnetic and sound waves.

Contextualization

Trigonometric functions describe phenomena that repeat periodically and, therefore, are essential in various practical applications. For example, they are used to understand and measure sound waves in physics and music. In engineering, they are used to describe oscillatory motion, for example, in bridges or buildings, and even in the design of gear wheels in mechanics.

Furthermore, trigonometric functions play a crucial role in the analysis of periodic data, such as economic cycles and weather patterns. Moreover, they are vital in the representation of images and sound in computer graphics and signal processing, respectively, such as in the compression and reconstruction of digital signals (JPEG, MP3, etc.).

Atividade Prática

Activity Title: Trigonometry in Real Life

Project Objective:

Explore and understand the application of trigonometric functions in real life, developing a practical and engaging experience with the aim of deepening knowledge on the subject and promoting skills such as teamwork, problem-solving, and critical thinking.

Detailed Project Description:

In this activity, groups will identify a real periodic phenomenon (such as sound waves, cyclical movements, phases of the moon, etc.) and model it using trigonometric functions. Students should research, explore, experiment, and present their findings through a presentation and a written report.

Required Materials:

Internet access for research, calculation application or algebra software (can be online or installed on a computer), paper and pencil for sketches and notes, camera for documentation (optional).

Step by Step:

1. Form groups of 3 to 5 students and choose a periodic phenomenon to analyze.

2. Conduct preliminary research to understand the chosen phenomenon. At this point, it is not necessary to address mathematical modeling, just to understand the general characteristics of the phenomenon.

3. Explore how trigonometric functions relate to the chosen phenomenon. This may involve the use of mathematical software to manipulate equations and graphs, practical experiments, or a combination of both.

4. Document the entire process, including notes, sketches, photos, graphs, etc.

5. Prepare a presentation of the research and findings. The presentation should be visually appealing, informative, and engaging in a way that is understandable to your peers who did not participate in the investigation.

6. Write a detailed report covering the research work, exploration of trigonometric functions, experimentation, and conclusions. The guidelines for writing the report are explained below.

Project Delivery:

Each group must deliver:

1. The project presentation, which can be done in slides or video.

2. A written report including the following topics:

   • Introduction: Describe the chosen phenomenon and why it was selected. Explain the relevance and application of the phenomenon in the real world and the role of trigonometric functions in its representation.

   • Development: Detail the research conducted, the trigonometric function used, the experimentation process, and the findings. Present graphs, value tables, photographs, or other materials that help understand the work done. A detailed discussion of the results is expected in this section.

   • Conclusion: Summarize the main points of the project, provide a summary of the learnings obtained, and analyze the effectiveness of modeling using trigonometric functions.

   • Bibliography: List all sources of information, such as books, websites, videos, etc., that were used during the project.

Students have one week to complete the project, and deliveries must be made in digital format. The presentation and report will be evaluated both for content mastery and for the quality of group work and presentation skills.