Introduction and Context

Introduction

Trigonometry, which is a branch of mathematics that studies the relationships between the sides and angles of triangles, has great relevance in high school and in various areas of knowledge. It allows us to analyze and solve problems related to angles and lengths in real-life situations. This project will focus on the basic concepts of trigonometry, specifically on sine (sin), cosine (cos), and tangent (tan).

These trigonometric functions are established from a right triangle, where the sine of an angle is the ratio between the length of the opposite side and the length of the hypotenuse; the cosine of an angle is the ratio between the adjacent side and the hypotenuse; and the tangent of an angle is the ratio between the opposite side and the adjacent side. These concepts form the basis of trigonometry and are extremely useful in a variety of fields.

Context

The application of trigonometry goes far beyond triangles and geometric figures, being an essential tool in various areas of knowledge. In engineering, trigonometric functions are used to calculate tensions and resistances in structures. In physics, they are applied to understand wave motion, such as that of a vibrating guitar string, being also fundamental in the understanding of light and other types of electromagnetic waves.

In navigation, trigonometry allows calculating straight-line distances between two points on a map. In biology, it helps in understanding growth patterns in spirals found in some organisms, such as mollusks and galaxies. In medicine, it is used in imaging diagnostics, such as tomographies and ultrasounds. These are just a few examples of how trigonometry is present in our lives, showing how important the study and understanding of these concepts are.

Practical Activity: "The Pythagorean Theorem in Practice and Trigonometric Functions"

Project Objective

This project aims to:

1. Encourage teamwork and cooperation;
2. Facilitate the understanding of the concepts of sine, cosine, and tangent;
3. Develop technical skills for calculating and solving trigonometric problems;
4. Stimulate time management and proactivity.

Project Description

Students will be grouped into teams of 3 to 5 members. They will work on a project that will involve building a model of a right triangle using simple materials such as cardboard, ruler, compass, and adhesive tape. Subsequently, they will take measurements and perform calculations to determine the trigonometric functions of the acute angle of the triangle, thus applying in practice the definitions of sine, cosine, and tangent.

The project should be completed within a month, and each student should dedicate between five to ten hours of work.

Required Materials

1. Cardboard
2. Ruler
3. Compass
4. Adhesive Tape
5. Pen or Marker
6. Calculator

Step by Step

1. Each group of students must build a model of a right triangle using cardboard. One of the angles should be 90 degrees (right angle), while the other angle should be acute and chosen by the group.

2. Once this is done, students should use the ruler to measure the sides of the triangle.

3. Students must calculate the length of each side of the triangle using the Pythagorean Theorem to confirm the validity of the model.

4. Using these measurements, students must calculate the trigonometric functions (sine, cosine, tangent) of the acute angle they chose.

5. Finally, students should compare their results with the trigonometric functions calculated using a calculator.

Student groups should document all steps of the process, from construction, measurement, calculations, and result comparisons, in a report format.

The report should include:

1. Introduction: Here students should explain what trigonometry is, its relevance, and how it is applied in the real world. Students should also describe the objective of this project.

2. Development: Students should detail the steps they followed to carry out the project, explain the theoretical concepts of sine, cosine, and tangent, as well as the methodology used. Students should present and discuss the results obtained.

3. Conclusion: Students should conclude the work by summarizing their main points, stating the learnings obtained, and drawing conclusions about the project.

4. Bibliography: Students should indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Remember that the report writing should be coherent with the work done, showing clearly and precisely the entire process the group went through to reach the final results.