Contextualization

Trigonometric functions are undoubtedly one of the key concepts in mathematics, with a wide range of applications in different domains, both within and outside of mathematics. At the heart of these functions are the sine, cosine, and tangent, which you have probably encountered before. These are periodic functions that describe the movements and shapes of waves, circles, and triangles, among other things.

To understand the depth of trigonometric functions, we first need to understand the concept of an angle, particularly in the context of a unit circle. Consequently, we will learn about the sine and cosine of an angle, as well as the tangent, which is defined in terms of both. With this knowledge, we will be able to explore how these functions behave, how they relate to geometry, and what they can tell us about the world around us.

Throughout this project, you will further explore the role of sine, cosine, and tangent, as well as their trigonometric relationships, such as the Pythagorean identities. Additionally, you will also examine and understand the values of sine, cosine, and tangent, and the graphs of these functions, which will allow you to see the periodicity of these functions and also the maximum and minimum values they can reach.

Trigonometric functions have a variety of real-world applications. For example, they are used in physics to understand waves and oscillations, in engineering to analyze and design structures, and in computer science for signal processing and graphics. Furthermore, trigonometric relationships can be seen in natural patterns, such as the cycles of the moon and the seasons of the year.

As such, a solid understanding of trigonometric functions and their values is not only relevant to the study of mathematics, but also provides a foundation for a variety of disciplines and careers.

To delve deeper into the study, we suggest the following resources:

• "Trigonometric Functions: Trigonometric Circle, sine, cosine, and tangent functions" - This comprehensive resource provides an accessible introduction to trigonometric functions, including the trigonometric circle and how to derive the sine, cosine, and tangent functions.
• "Definitions of sine, cosine, and tangent" - A detailed explanation of the definitions and relationships of sine, cosine, and tangent, with clear examples.
• "Trigonometric Functions: What they are, formulas, examples, and exercises" - This is a comprehensive guide on trigonometric functions, their formulas, and solved examples.
• "Trigonometric Functions" - This site offers a detailed explanation of trigonometric functions, as well as interactive graphs to help visualize how these functions work.

Practical Activity

Activity Title: "Trigonometry in the Real World: From Theory to Practice"

Project Objective:

This project aims not only to improve students' technical knowledge of Trigonometric Functions but also to enhance socio-emotional skills, such as teamwork, time management, and problem-solving.

Detailed Project Description:

1. Theoretical research and presentation: Initially, each group will conduct comprehensive research on Trigonometric Functions, focusing on the topics of sine, cosine, tangent, the relationships between them, and the values of these functions. Each group will make a theoretical presentation in slides explaining these concepts to the rest of the class.

2. Exploration of real-world applications: Next, the groups will identify and explore real-world applications of Trigonometric Functions, such as in the study of natural cycles, sound waves, structural engineering, etc. Each group will choose a specific application to focus on and make a presentation explaining how trigonometry applies to that area.

3. Creation of a practical project: After the research and presentation, each group will develop a practical project to demonstrate the application of Trigonometric Functions in the real world. It can be an experiment, a simulation, a prototype, a case study, or any other relevant idea. The project should include the use of at least two disciplines, such as physics and mathematics, for example.

4. Documentation and Project Presentation: Each group must document the entire development process of their project in a detailed report that will be presented to the class, including the challenges encountered and how they were resolved, the methodology used, and the results or conclusions obtained.

Required Materials:

1. Computer with Internet access for research and presentation.
2. Various materials depending on the chosen practical project (everything must be safe and approved by the teacher).

Detailed Step-by-Step Guide for the Activity:

1. Students are divided into groups of 3 to 5 people.
2. Each group conducts research on Trigonometric Functions and identifies a specific real-world application.
3. Each group creates a slide presentation explaining the concepts of Trigonometric Functions and the chosen application.
4. The groups present their findings and applications to the rest of the class.
5. Each group plans and executes a practical project based on their chosen application.
6. During the project execution, the groups document their process, challenges, solutions, and results in a report.
7. The groups present their projects and reports to the class and discuss their conclusions.
8. The teacher evaluates each group based on the quality of the presentations, the project and the report, as well as teamwork and participation.

Project Deliverables:

1. Presentations: Each group must make two presentations to the class - one to explain the concepts of Trigonometric Functions, and one to explain their chosen application.

2. Practical Projects: Each group must develop a practical project demonstrating the application of Trigonometric Functions in the real world.

3. Reports: Each group must submit a detailed report documenting their project, including an introduction to the topic, description of the methodology used, discussion of the results obtained, and a conclusion summarizing their findings and lessons learned. The report should also include a bibliography of the sources consulted.

Remember: the process of writing the report should not be seen as a separate task, but as a continuous part of the project. Throughout the process, students should take notes, record their processes, methods, and findings, which will be useful when compiling the final report.