Context

The Law of Sines is one of the most important concepts in trigonometry, a subfield of mathematics that studies the relationships between the sides and angles of triangles. The Law of Sines is applicable to all types of triangles, whether they are acute, obtuse, or right, and it allows us to solve many geometric problems. It is based on the idea that in any triangle, the ratio of the length of a side to the sine of the angle opposite that side is constant for all three sides.

Consider a triangle with sides of lengths a, b, and c and angles opposite those sides of measures A, B, and C respectively, the Law of Sines states that:

a/sin(A) = b/sin(B) = c/sin(C)

This law is a powerful tool in fields such as physics, engineering, and architecture, as well as many other areas where we need to calculate distances and angles.

Have you ever wondered how engineers calculate the height of a building, or how sailors determine the distance between two points in the ocean? The Law of Sines is often used in these situations and many more.

Understanding why this law works and knowing how to apply it to solve practical problems is a valuable and useful skill, not only for school exams and tests, but also for a wide variety of careers and everyday situations.

The following resources can be used as references for further study on the topic:

1. Khan Academy - Law of Sines
2. Brilliant - Law of Sines and Cosines
3. Math is Fun - Law of Sines

Hands-on Activity: Solving Mysterious Triangles

Project Goal

To explore the Law of Sines through the study and solving of practical problems, while developing essential skills such as teamwork, communication, time management, and problem-solving.

Project Description

Each group will take on the role of mathematical detectives to solve a series of challenges that involve using the Law of Sines. Each challenge consists of solving a "case", which is a real-world practical problem in which the Law of Sines is vital to finding the solution.

Materials Required

• Graph paper
• Ruler
• Compass
• Scientific calculator
• Internet access
• Computer or laptop

Step-by-Step Instructions

1. Each group will be given a set of "cases" (geometric problems that require the Law of Sines to be solved). The cases can range from finding the distance between two points on a sea route to calculating the height of an unknown building.

2. The group should discuss and plan how to solve each case, making use of the available materials, the knowledge acquired in class, and external research sources if necessary.

3. The group should then work together to solve the cases, carefully documenting each step of the process in a logbook, explaining the decisions made, the theory used, the calculations performed, and presenting the results obtained.

4. Once all the cases have been solved, the group should then produce a final report, containing the following items:

   • Introduction: Background on the Law of Sines, its relevance in the real world, the goal of the project, and a summary of the cases faced.
   • Development: Presentation of the theory of the Law of Sines and its application in solving the cases. Details of the methodology used, discussion of the problem-solving processes for each case, and presentation of the final results.
   • Conclusion: Overall reflection on the work, main learnings, challenges faced and overcome, and the importance of teamwork in solving the cases.
   • Bibliography: Sources used to support the project, such as books, websites, videos, and other resources that the group found useful.

5. The final report should be submitted in digital format, respecting the maximum page limit established. In addition, one member of the group should give an oral presentation of the work to the class, highlighting the main challenges faced in the cases and how the Law of Sines was used to solve them.

It is estimated that this project will take approximately two weeks to complete, with each student dedicating a minimum of 12 hours of individual work throughout this period.