Contextualization

Trigonometry is the area of Mathematics that studies the relationships between angles and the measures of the sides of a triangle. These relationships are expressed through the so-called trigonometric ratios, which are of fundamental importance for both practical applications and for more advanced studies in mathematics, physics, engineering, among others.

The main trigonometric ratios analyzed in a right triangle are: sine (sin), cosine (cos), and tangent (tan). These ratios are defined in terms of an acute angle chosen in the right triangle and are related to the measures of the sides of that triangle. For specific angles, such as 30º, 45º, and 60º, these ratios have known values that are widely used in calculations.

Trigonometry plays a crucial role in areas as diverse as navigation, engineering, physics, computer graphics, and even music! In fact, any situation that involves length measurements and angles can benefit from the use of trigonometry. A practical example is the use of trigonometry in solving GPS location problems, which require precise measurements of distances and directions.

In more abstract terms, trigonometry helps to understand and describe periodic phenomena, such as the movement of planets around the sun or the behavior of light and sound waves. Without trigonometry, many of the technological advances we have today would not be possible.

For the elaboration of the suggested work, the following sources are recommended for consultation and deepening:

1. Trigonometry in the right triangle - Brasil Escola
2. Trigonometric ratios - Mundo Educação
3. The Trigonometry of the Right Triangle - Só Matemática
4. Trigonometry - Khan Academy

Practical Activity

Activity Title: "Finding the Way with Trigonometry"

Project Objective:

By executing this project, students will be able to understand and apply the main trigonometric ratios (sine, cosine, and tangent) in the context of a practical problem of location and navigation. The goal is for them to understand the importance of these ratios not only in a theoretical context, but mainly in their practical applications.

Detailed Project Description:

Students will be divided into groups of 3 to 5 members. Each group will be tasked with creating a "navigation route" using angles and trigonometric ratios. The challenge is to establish a sequence of instructions to go from one point to another on a flat map (such as a map of a park or a school/university campus) using only distance and direction information provided by trigonometric ratios.

Necessary Materials:

1. Millimeter paper or grid paper;
2. Ruler;
3. Compass;
4. Scientific calculator;
5. Pencil and eraser.

Detailed Step-by-Step for Activity Execution:

1. Firstly, the group must choose or create a flat map for their project. It can be a real or fictional map, as long as it has clear reference points (such as buildings, trees, or any other clearly identifiable object).
2. The group must then choose or determine two points on the map, a starting point and an endpoint.
3. From the starting point, the group must trace a route to the endpoint. This route does not need to be the shortest, the goal is precisely to create a path that involves changes of direction, which can be expressed through angles.
4. Now is the time to apply trigonometry. For each segment of the route, the group must calculate the measures of the sides of an imaginary right triangle, where the hypotenuse is the path to be traveled, one of the angles is the course to follow (in relation to some fixed direction, such as north, for example) and the catheti represent the distances to be traveled in the east-west and north-south directions.
5. The measures of the angles and catheti will then be translated into navigation instructions, such as "go X meters east, then Y meters north", etc. Here, students must use trigonometric ratios to calculate the distances to be traveled.
6. Finally, the complete route, expressed in terms of navigation instructions, must be documented in a report, which will be delivered together with the map used.

At the end of the project, students must conclude the project report with the following components:

Introduction

In this section, students must explain the reason for the project, the relevance of trigonometry, and its application in everyday life. The worked problem and the goal to be achieved should be contextualized.

Development

Here, students must explain the theory behind the trigonometric ratios and how these were applied in practice. The step-by-step of the activity must be reported in detail, including the difficulties encountered during the project execution and how these were overcome. The measurements, calculations, and navigation instructions must be clearly presented and justified. Group discussions and decisions made should also be recorded.

Conclusion

The main learnings obtained should be highlighted, how teamwork contributed to the project, and the importance of trigonometry in practical life. Suggestions for improvements or modifications to the project can also be included.

Bibliography

The learning resources used by the group must be properly cited, following ABNT standards.

This practical activity, together with the elaboration of the report, is estimated to take between five to ten hours per student to be executed and has a deadline of one month after the start date.