Contextualization

Trigonometry plays a fundamental role in mathematics and its applications in physical sciences, engineering, econometrics, and other areas. One of the primary tools of trigonometry is the concept of multiple angles, particularly double and triple angles. We will delve into their study in this project since this topic allows for a variety of real-world applications, ranging from wave modeling to satellite navigation.

The formulas for double and triple angles for sine, cosine, and tangent functions generalize and extend the periodicity of these functions. For example, the sines and cosines of angles of 2x, 3x, 4x, etc., allow us to explore and solve complex problems involving circular motion, harmonic waves, and even frequency issues in the field of electrical engineering.

Studying the double and triple of angles in terms of sine, cosine, and tangent not only deepens your knowledge of trigonometry, but it also helps to improve your problem-solving skills for real-world problems. These concepts are essential for solving more complex trigonometric equations. Solving problems involving these equations will improve your analytical thinking skills and will also increase your confidence when dealing with more advanced mathematical problems.

Resources that can be used to deepen the understanding of this topic:

1. Book: "Fundamentos de Matemática Elementar: Volume 3 – Trigonometria" by Gelson Iezzi and Carlos Murakami. Publisher: Atual Editora
2. Math Website: Só Matemática (https://www.somatematica.com.br) is an educational portal containing a large amount of content on trigonometry, including the concepts of double and triple angle.
3. Video lessons: YouTube math channel "Ferretto Matemática" (https://www.youtube.com/user/ProfessorFerretto/featured)

Practical Activity

Activity title: Unveiling the World of Multiple Angles

Project goal:

To deeply understand the formulas of double and triple angles of sine, cosine, and tangent and to apply them to real-world problems.

Detailed description of the project:

In this project, students divided into groups of 3 to 5, will be challenged to investigate the concept of double and triple angles in trigonometry and to demonstrate how they can be applied in solving real-world problems. Besides strengthening the theoretical concepts, the activity will encourage collaboration among students since teamwork will be essential to the success of the project.

Students will be encouraged to use digital resources (e.g. mathematical software) to aid in the solving of proposed problems and in the graphical representation of the trigonometric functions under study.

The project will be carried out in four stages:

Stage 1: Theory

Students should start with the theoretical study of double and triple angles. For this, they should use the suggested book and supplement the study with web resources and suggested videos. In this stage, the group must review the concepts of sine, cosine, and tangent and then delve into the formulas that involve double and triple angles.

Stage 2: Mathematical Challenges

After the theoretical study, the group will have to solve a set of problems involving the sine, cosine, and tangent of multiple angles. It is important to document the resolutions of each problem for future reference.

Stage 3: Application in a real-world problem

Now, students must choose a real-world problem that can be solved using the concepts learned. The problem can be from any field that involves mathematics, such as physics, engineering, or economics. After selecting the problem, the students must solve it and document the whole process.

Stage 4: Demonstration and explanation

Finally, the group must create a presentation that contains the explanation of the studied theoretical concepts, the problem resolutions, and, mainly, the application of the concepts to a real-world problem.

Materials required:

• Book "Fundamentos de Matemática Elementar: Volume 3 – Trigonometria" by Gelson Iezzi and Carlos Murakami
• Access to the internet (for consulting and using relevant digital resources)
• Math software or scientific calculators

Project deliverables:

At the end of the project, each group will have to submit a detailed report containing:

1. Introduction: Description of the theme, its importance, and application in the real world, and project goal.
2. Development: Description of the theory, the activity in detail, the used methodology, presentation of the obtained results, and a discussion of those results.
3. Conclusion: Review of the project's main points, learned lessons, and the drawn conclusions of the project.
4. Bibliography: Reference sources that were used for the project, such as books, websites, videos, etc.

The presentation of stage 4 (demonstration, and explanation) can be made in slide format and will be a fundamental part of the report.