Contextualization

Trigonometry, a word that comes from the Greek "Trigōnon", which means "triangle", and "Metrēo", which means "measure", is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is an essential tool in various areas of knowledge, such as physics, engineering, geography, architecture, and even in music!

The fundamental concepts of trigonometry are sine, cosine, and tangent. These are mathematical functions that relate an angle of a right triangle to the proportions between its sides. The sine of an angle is the ratio between the side opposite that angle and the hypotenuse. The cosine is the ratio between the adjacent side and the hypotenuse. The tangent, in turn, is the ratio between the opposite side and the adjacent side.

Learning to understand and apply these relationships is fundamental for the development of logical-mathematical skills and thinking. This knowledge will open doors to understanding and solving more complex problems in various disciplines and everyday situations.

An example of the application of trigonometry in the real world is in architecture. Have you ever noticed how the pyramids of Egypt have a specific inclination? This angle was thought out using principles of trigonometry. This is just one of the many ways in which Trigonometry is present in our daily lives.

The basis for this project will be the book "Fundamentals of Elementary Mathematics: Trigonometry - 9th Ed. 2012" by Gelson Iezzi and Carlos Murakami. We also recommend the use of the website of "Professor Ferreto", which offers a series of detailed video lessons on the subject. For practice and exercise solving, we recommend "SOS Professor: Trigonometry", an online platform that allows step-by-step problem solving.

Project Introduction

In this project, you will deepen your knowledge of sine, cosine, and tangent, understanding their relationships and applications. All of this will be done in a practical and playful way, through activities that engage and motivate learning. The work will be carried out in groups, and you will have the opportunity to develop socio-emotional skills such as communication, problem-solving, creative thinking, proactivity, among others.

We hope that by the end of the project, you will be able to understand and apply the concepts of sine, cosine, and tangent in everyday situations, and see how mathematics is a powerful tool for understanding the world around us.

Practical Activity: "Trigonometry in the Real World: The Fuse of a Firework"

Project Objective

This activity aims to apply the concepts of sine, cosine, and tangent in a practical and real scenario: the trajectory of a Firework. Students will explore how trigonometry can help understand and predict the behavior of an object in motion.

Detailed Project Description

Groups will be challenged to simulate the trajectory of a firework using trigonometry concepts. The idea is to determine the launch angle for the firework to reach a specific point. The project will be divided into two parts:

1. In the first part, groups must study and research the concepts of sine, cosine, and tangent, and how they can be applied to determine the trajectory of a firework.

2. In the second part, groups must use these concepts to simulate the firework's trajectory. This can be done on a piece of paper, with the help of digital drawing or programming tools, or even using robotics and programming kits, if available.

Teams will have coordination to conduct research, make calculations, and prepare presentations, thus challenging their collaborative, communication, and problem-solving skills.

Required Materials

• Trigonometry books, especially "Fundamentals of Elementary Mathematics: Trigonometry - 9th Ed. 2012"
• Internet access for complementary research and video lessons (such as those from "Professor Ferreto")
• Sheets of paper, pencils, erasers, and rulers for drawing
• Simulation software, such as GeoGebra, or robotics kits, if available
• Equipment for recording videos or presentations, such as a smartphone or computer.

Detailed Step-by-Step for Activity Execution

1. Review and research of the concepts of sine, cosine, and tangent, focusing on their application in physics (projectiles and parabolic motion).

2. Group discussion on how to apply these concepts to predict the trajectory of a firework.

3. Choose the simulation method: drawing on paper, using software, or building with robotics kits.

4. Drawing or building the simulation, determining the launch angle of the firework.

5. Performing calculations and adjustments for the firework to reach the intended point.

6. Preparation of the presentation, explaining the research process, drawing, and construction, and demonstrating the simulation.

7. Presentation to the class, followed by a question and answer session.

8. Writing a detailed report, explaining all the steps followed, the conclusions reached, and the bibliography used.

Project Deliverables

Students will have two main deliverables:

1. Presentation of the simulation to the class, explaining the calculations performed and the concepts used.

2. The written report containing all the project steps, from the initial research to the conclusion. As well as an introduction explaining the relevance of Trigonometry, a detailed description of the practical activity, the presentation and discussion of the results obtained, the conclusions, and the bibliography used.

The report should be written clearly and concisely, showing the students' understanding of Trigonometry concepts and their application in solving real problems.

By the end of the project, students will be more familiar with sine, cosine, and tangent, and better equipped to apply these concepts in other areas of mathematics and everyday life.