Contextualization

Trigonometry is one of the most fascinating and practical branches of Mathematics, ranging from calculations in civil construction to tide prediction and video game programming. In this project, we will focus on understanding and applying the basic trigonometric lines: sine, cosine, and tangent of the angles 30º, 45º, and 60º.

Mathematics is the language of reality. Whether in engineering, physics, chemistry, economics, or even in the arts, mathematics is an essential tool for understanding and transforming the world. Specifically, Trigonometry, an area that deals with the study of relationships between angles and distances, is an indispensable part of this. Understanding the basic trigonometric lines and how they apply to everyday situations is the first step to acquiring fluency in this language.

Introduction

Trigonometry originated from ancient studies of the stars and their movement. It became the basis for precise measurement of distances and angles. And the basic trigonometric lines - sine, cosine, and tangent - are the fundamental tools of this field of knowledge.

Sine, cosine, and tangent are, essentially, ratios between sides in a right triangle. They are called trigonometric functions because from them, it is possible to calculate angles, sides, and areas of figures. They are so crucial to mathematics that they are the foundations for all of trigonometry.

For the angles of 30º, 45º, and 60º, sine, cosine, and tangent have values that are easily calculable and, therefore, provide an excellent starting point for the study of trigonometry. Knowing them and how to obtain them will open doors to other areas of mathematics and to a multitude of applications.

Practical Activity

Activity Title: 3D Trigonometric Model

Project Objective

The objective of this activity is to provide a practical and theoretical understanding of the basic trigonometric lines (sine, cosine, and tangent) for the angles 30º, 45º, and 60º, and how these concepts are applied in the real world.

Detailed Project Description

Students will create a 3D model of a set of right triangles with angles of 30º, 45º, and 60º. They will also calculate the values of sine, cosine, and tangent for these angles and verify their observations with the help of the model. They will then analyze and discuss their findings in a written report.

Required Materials

• Cardboard
• Ruler
• Protractor
• Colored pens
• Calculator

Detailed Step-by-Step

Step 1: Divide students into groups of 3 to 5.

Step 2: Each group should create a set of right triangles with angles of 30º, 45º, and 60º using cardboard, ruler, and protractor. The measurements of the triangle sides should be clearly marked.

Step 3: Use the calculator to calculate the value of sine, cosine, and tangent for each of the angles. Record these values.

Step 4: Verify your observations using the model. For example, the sine of an angle is the ratio between the opposite side and the hypotenuse. Check if this is true for your model.

Step 5: Analyze and discuss the findings. What were the observations? Was there any surprise? How do the results fit into the theory you know?

Step 6: Each group should produce a written report detailing their findings. The report should follow the following structure:

1. Introduction: Contextualize the project, explain its relevance and purpose, and briefly discuss the use of trigonometry in the real world.
2. Development: Write a theoretical review of the basic trigonometric lines and the angles used in the project. Describe the activity carried out, the methodology used, and discuss the results based on the theory.
3. Conclusion: Reflect on the lessons learned, the challenges, and surprises during the project. Report the main points of the project and the main conclusions.
4. Bibliography: Indicate all sources used for the project: books, websites, videos, etc.

Project Deliverables

The student will deliver the 3D model of a set of right triangles and a detailed written report. The 3D model will serve as a visual and practical confirmation of the reflections described in the report. The report will detail the steps of the activity, the relevant theory, the observations made, the findings, and the conclusions obtained during the activity. The report should connect to the 3D model, using it as a reference to validate the findings and conclusions.