Contextualization

Theoretical Introduction

Trigonometry is an area of Mathematics that studies the properties of trigonometric functions, especially sine, cosine, and tangent. Trigonometric functions are based on the relationship between angles and measurements in triangles. In particular, in unit circles, these functions are extremely important in various practical applications.

The graphs of trigonometric functions consist of visual representations of the sine, cosine, and tangent functions. They are oscillatory and periodic, which means they repeat at regular intervals. This characteristic is fundamental for understanding natural phenomena that follow repetitive patterns, such as moon phases, ocean tides, sound waves, among others.

To fully understand these functions, it is essential to understand their graphical behavior. The study of graphs allows not only the visualization of functions but also the possibility of predicting future behaviors based on past patterns. Furthermore, knowledge of a function's graph facilitates the resolution of trigonometric equations.

Contextualization

In the real world, trigonometric functions have essential applications in various areas. In Physics, for example, they are used to analyze wave phenomena, such as sound and light. In Engineering, they are applied to calculate forces in inclined structures. In Astronomy, they help in understanding planetary orbits.

Moreover, in our daily lives, we find applications of trigonometric functions in various situations. Our clock, for example, which has periodic movement, is a physical representation of the cosine function. The sound perceived in a musical show is a wave, whose behavior can be modeled by the sine and cosine functions. Understanding the graphs of these functions allows us to better understand the world around us.

Therefore, understanding trigonometric functions and their graphical representations is essential not only for learning Mathematics but also for the comprehension and interpretation of a large part of the natural and technological world.

Practical Activity

Activity Title: "Dancing with Trigonometric Functions"

Project Objective

This activity aims to familiarize students with the graphs of trigonometric functions and their application in interpreting real periodic phenomena. We will work especially with the sine and cosine functions.

Students will be challenged to create a "dance of trigonometric functions," that is, a choreography that visually represents the graphs of the sine and cosine functions.

Detailed Project Description

In groups of 3 to 5 members, students will explore the fundamental principles of trigonometric functions and their graphs. The idea is for them to translate these abstract concepts into a more practical and playful medium through the creation of a dance choreography.

Each group member will be responsible for representing one of the trigonometric functions (sine and cosine) or a time marker (indicating angles from 0° to 360°). The students representing the functions should move in a way that their movements correspond to the value of the function at each angle (for example, at the highest point of the movement for the maximum value of the function, at the lowest point for the minimum value).

This project should be carried out in two parts: the first part will be dedicated to the theoretical understanding of the functions and the creation of the choreography, and the second part will be dedicated to the practice of the dance and the video recording.

Required Materials

• Blackboard or whiteboard for theoretical explanation.
• Paper and pens or pencils for notes and drawings.
• Ample space for dance practice.
• A camera or cell phone for video recording.

Detailed Step-by-Step

1. Study of trigonometric functions: Students should review the sine and cosine functions and their graphs. They must have a clear understanding of the behavior of these functions throughout a cycle (0° to 360°).

2. Choreography creation: Students should create a choreography representing the graphs of the functions. It is important that they can adequately portray the peak and valley moments of the functions.

3. Practice and recording: After creating the choreography, students should practice it and then record the dance. The recording should be done in a spacious place with good lighting. One of the group members should be responsible for recording the video.

4. Report preparation: Finally, students should prepare the project report, detailing the entire process of creating and executing the dance. The report should include the sections of introduction, development, conclusions, and bibliography used.

Project Deliverables

At the end of the project, each group must deliver:

1. A video, with a duration of up to 5 minutes, presenting the "Dance of Trigonometric Functions." The video should be clear and have good image and sound quality. Each movement of the dancers must be clearly associated with a specific point on the function graph they are representing.

2. A written report, with a maximum of 10 pages, detailing the experience. The report should include:

• Introduction: Contextualization of the theme, relevance of trigonometric functions and their graphs, project objective.
• Development: Detailed explanation of the sine and cosine function graphs, description of the activity execution (including choreography creation, dance practice, and video recording), discussion of the results obtained.
• Conclusions: Reflection on the learnings obtained, discussion on the effectiveness of the playful method used for learning trigonometric functions and their graphical representations.
• Bibliography: References of the information sources used for the project realization.

The report should be written clearly, cohesively, and coherently, and should demonstrate the students' understanding of the theme and the activity carried out.