Contextualization

Considered one of the most important trigonometric relationships, the Law of Cosines has a wide range of practical applications in various areas of knowledge, such as Physics, Engineering, and even Computer Science. Whether it's to calculate the depth of a lake without the need to dive, estimate the distance between two points on a GPS, or determine the intensity of forces in equilibrium, the Law of Cosines is present and crucial.

The Law of Cosines, essentially, is an extension of the Pythagorean Theorem for triangles that are not necessarily right-angled. It is based on the relationship between the sides of a triangle and the cosine of one of its angles. Exactly for this reason, this law is a powerful tool for solving problems involving the determination of sides and angles of a triangle.

To support the study and deepening of the topic, students can use the following didactic resources:

1. Book: "Baldor, A. (2014). Geometria Plana e Álgebra. Translated by M. M. R. Pombo. Rio de Janeiro: Harbra."
2. Video: Matemática Rio Channel with Prof. Rafael Procópio, "Law of Cosines - Very Easy". Available at: Matemática Rio - Law of Cosines
3. Website: Mundo Educação, "Law of Cosines". Available at: Mundo Educação

These resources were selected for their quality, accuracy, and ease of understanding and should provide a solid understanding of the basic concepts of the Law of Cosines and some of its most popular applications.

Practical Activity: "Deciphering the Law of Cosines through the Space Sideral"

Project Objective

This project aims to apply the Law of Cosines to model and solve problem situations. The project also aims to develop socio-emotional skills such as time management, communication, problem-solving, creative thinking, and proactivity.

Detailed Project Description

Students will be divided into groups of three to five people, and each group will be tasked with creating a three-dimensional model of a planetary system. The planets will be represented by styrofoam spheres, and the radii of the orbits will be measured and calculated using the Law of Cosines.

Necessary Materials

1. Styrofoam spheres of different sizes
2. Nylon threads for the orbit radii
3. Ruler
4. Scientific calculator
5. Pencils, erasers, and paper for calculations
6. Paint and brushes for the decoration of the planetary system

Detailed Step-by-Step for Carrying Out the Activity

1. Research and Planning: Each group should research a specific planetary system and plan how to represent it three-dimensionally.

2. Model Construction: Using the styrofoam spheres, students will build a physical model of the chosen planetary system. The orbits of the planets will be represented by nylon threads.

3. Measurement and Calculations: With the help of a ruler and a scientific calculator, students will measure the radii of the orbits and use the Law of Cosines to calculate the distance between any two planets in a specific configuration.

4. Activity Report: Finally, each group will write a report describing the entire process, with an emphasis on the application of the Law of Cosines. This report should include the following topics:

   • Introduction: The student should contextualize the theme, its relevance and application in the real world as well as the objective of this project.
   • Development: The student should explain the theory behind the central theme(s) of the project, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: The student should conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
   • Bibliography: The student should indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

The report must be submitted one week after the presentation of the practical activity to allow adequate time for writing.