Contextualization

Our journey begins with trigonometry, a fundamental branch of mathematics that deals with the relationships between angle measures and distances between points. There is a vast application for trigonometry, especially in physics, engineering, computer science, and many other areas.

We will come into contact with three basic trigonometric functions: sine, cosine, and tangent. These functions are defined from a unit circle (a circle with a radius equal to 1) concerning an angle. We will specifically examine the angles of 30°, 45°, and 60°, which have unique and important properties.

These functions are the basis of trigonometric identities, which are equations that relate the trigonometric functions to each other. They are essential for solving many practical and mathematical problems.

Importance and real-world application

Trigonometry is a relevant field of study not only for mathematics but also for many other disciplines and practical applications. In physics, for example, it is used to study waves, circular and oscillatory motion, among other phenomena.

In everyday life, trigonometry has many applications. It is used in navigation to calculate distances and routes; in civil engineering to design structures; and even in music to analyze and produce sounds.

Understanding trigonometric functions and how they relate to angles and triangles will help us better understand these practical applications, as well as open doors to advanced fields of study.

I recommend the following resources for further study on the topic:

• [Khan Academy] (https://pt.khanacademy.org/math/trigonometry) - Complete trigonometry course with explanatory videos.
• [Calcule Mais] (https://www.youtube.com/watch?v=ppLOgBmF2MY&list=PLIEQ9jySJHo6xm5duoAwQJv0vhCOL6RLm) - YouTube video series on trigonometry.
• [IME-USP] (https://educapes.capes.gov.br/bitstream/capes/68636/2/Trigonometria%20no%20ensino%20m%C3%A9dio.pdf) - Digital book on high school trigonometry, with theory and solved exercises.

Practical Activity - "Conquering the Trigonometric Triangle"

Project Objective

The main objective of this project is to deepen the understanding of basic trigonometric functions by exploring the angles of 30º, 45º, and 60º. Additionally, it is expected that students will develop skills such as teamwork, time management, and the ability to apply theory to practice.

Project Description

The activity is in groups of 3 to 5 students and consists of constructing three three-dimensional models of right triangles, where the non-right angles will be 30º, 45º, and 60º. Subsequently, students should measure the sides of each triangle, calculate the values of sine, cosine, and tangent of the angles, and compare them with theoretical values, proving the trigonometric relationships.

Necessary Materials

• Skewers or wooden sticks
• Glue
• Sewing thread or nylon thread
• Protractor
• Ruler
• Calculator

Step by step

1. Organize into groups of 3 to 5 students.
2. With the available materials, construct three three-dimensional structures of right triangles. Each triangle should have one of the angles (other than the right angle) being 30º, 45º, and 60º. Use the protractor to ensure the accuracy of the angles.
3. Measure the lengths of the sides of the triangles with the ruler and note them down.
4. Calculate the sine, cosine, and tangent of the angles of each triangle using the obtained measurements and verify if the values correspond to the theoretical ones.
5. Discuss possible errors or deviations and how they might have occurred.

Project Submission and Written Document

After completing the practical part, each group should produce a report on the activity. The report should include the following topics:

1. Introduction: Contextualization about trigonometry and justification for choosing the topic, as well as its relevance and real-world application.
2. Development: Detail the theory behind the basic trigonometric functions, explain the activity carried out step by step, present the methodology used, and discuss the obtained results.
3. Conclusion: Summarize the main points of the work, discuss the lessons learned, and the conclusions about the project.
4. Bibliography: Indicate the sources used as the basis for the project work.

Remember that the preparation of the report is as important as the construction of the triangles and should demonstrate the understanding of the concepts worked on. This document should be submitted along with the three-dimensional structures of the triangles.