Contextualization

In the study of mathematics, especially trigonometry, we often encounter situations where we need to calculate trigonometric lines for various angles. We often come across angles such as 45º, 30º, and 60º, which are quite common. However, there are some angles that may seem a little more complex at first glance, but which are of utmost importance and can be easily calculated with a few tricks. These are the angles of 75º, 15º, and 22.5º.

In trigonometry, trigonometric lines are represented by the sine, cosine, and tangent. These functions are used to relate the angles to the sides of a right triangle and are powerful tools for solving various geometric and physical problems.

Angles of 75º, 15º, and 22.5º are frequently found in everyday activities and nature. The geometric shapes formed by these angles are very common, such as the regular pentagon (which has interior angles of 75º) and some flowers and shells (which have angles of 15º and 22.5º in their formation). Furthermore, these angles also have fundamental application in engineering, architecture, and physics.

Introduction

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. For acute angles, these relationships are expressed by three trigonometric functions known as sine (sin), cosine (cos), and tangent (tan).

Calculating these trigonometric lines for angles of 75º, 15º, 22.5º may seem complex, but with the use of some well-known properties and with practice, it is a simple and quick task to perform.

To support this study, we suggest reading the following materials in Portuguese:

1. Book: "Trigonometry and Complex Numbers", by Dante Claramonte Deliberador (ISBN-13: 978-8521616569)
2. Educational website Khan Academy, with explanations and exercises on trigonometry: Trigonometry
3. Explanatory videos from the channel "Matemática Rio com Prof Rafael Procopio" on YouTube: Matemática Rio

Practical Activity: "Unveiling Angles and Trigonometric Lines"

Project Objective

The objective of this project is for students to become familiar with the calculation processes of the trigonometric lines sine, cosine, and tangent for the angle measurements of 75º, 15º, and 22.5º. Students will learn to use such calculations to solve real-world problems, perceive the application of these concepts in everyday life, improve their teamwork and communication skills, and reinforce their research, interpretation, and textual production skills.

Detailed Project Description

Students will be divided into groups of 3 to 5 members. Each group will be assigned the task of studying and calculating the trigonometric lines of one of the proposed angles (75º, 15º, 22.5º). Students should do research and put together a detailed report including the theory and practice of the concepts learned.

Required Materials

• Notebook or paper for notes
• Pens and pencils
• Ruler and compass
• Scientific calculator
• Internet access for research

Step-by-Step Instructions for Completing the Activity

1. Group formation: students will be divided into groups of 3 to 5 people, according to the total number of students in the class.

2. Angle division: each group will be assigned one of the angles (75º, 15º, 22.5º) to explore.

3. Researching: each group should research and learn how to calculate trigonometric lines for the assigned angle, using the materials provided.

4. Calculations: students should calculate the trigonometric lines sine, cosine and tangent for the assigned angle.

5. Application: each group should identify and solve a practical real-life problem that involves using the angle they studied.

6. Report preparation: the group should write a report with the collected information, calculations performed, and problem solution. The report must contain the following topics:

   • Introduction: Contextualization of the theme, its relevance and real-world application, as well as the objective of this project.
   • Development: Explain the theory behind calculating the trigonometric lines of the assigned angle, detail the activity carried out, indicate the methodology used, and present and discuss the results obtained.
   • Conclusion: Summarize the main points, clarify the lessons learned and the conclusions drawn about the project.
   • Bibliography: List the sources used to work on the project, such as books, web pages, videos, etc.

7. Presentation: each group will present the results of the study and the solution to the chosen problem to the class, explaining the concepts learned.

Project Deliverables

• Written report: each group must submit a complete report containing an introduction, development, conclusion, and bibliography. This report should detail the study carried out on calculating the trigonometric lines for the assigned angle, including the theory, methodology used, calculations performed, and solution to the practical problem chosen.
• Oral presentation: each group should prepare a 10-15 minute presentation describing the concepts learned, the methodology used and the results obtained.