Contextualization

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of a triangle. The name comes from Greek and means measurement of triangles. In short, we can say that this part of Mathematics studies triangles, especially their metric aspects, that is, the relationships between their measures. The basic concepts that guide the study of trigonometry include the sine (sin), cosine (cos), and tangent (tan) of an angle.

The sine of an angle in a right triangle is the ratio between the length of the side opposite the angle and the length of the hypotenuse. The cosine is the ratio between the length of the adjacent side and the hypotenuse. The tangent of an angle is the ratio between the sine and cosine of the same angle, or the ratio between the opposite side and the adjacent side.

The concepts of sine, cosine, and tangent are fundamental for understanding geometry and mathematics in general. These functions are present in various real-world applications, such as in physics, engineering, architecture, astronomy, among others. After learning these concepts in trigonometry, students will be prepared to understand and solve more complex problems involving the use of these functions.

Trigonometry is an essential component of Mathematics and has practical implications in various areas. Electromagnetic waves, for example, which are essential for data and TV transmission, are described using trigonometric functions. In physics, trigonometry is used to solve problems related to forces, sound waves, light, construction of structures, locomotion, and more. Even in music, trigonometry can be used to model sound waves.

Considering its importance and constant presence, both in academic and everyday life, it is essential that we master the basic concepts of trigonometry. Thus, we will be able to better understand the world around us and find solutions to complex problems. In addition, mastering trigonometry is a requirement for understanding more advanced content in mathematics, exact sciences, and engineering.

To delve deeper into the subject, you can consult the following sources:

• Trigonometry on Khan Academy
• Trigonometry on Só Matemática
• Trigonometry at the University of São Paulo (USP)

Practical Activity: "Building a Trigonometric Clock"

Project Objective

The activity aims to address the concepts of sine, cosine, and tangent in a practical and playful way, through the construction of a trigonometric clock. The project aims to stimulate teamwork, mathematical reasoning, and understanding the application of these concepts in daily life.

Detailed Project Description

In this activity, students will be divided into groups of 3 to 5 members and must build a trigonometric clock using recyclable materials. In this clock, the sine, cosine, and tangent functions will be represented by a system with three hands: one for sine, another for cosine, and the third for tangent. The goal is for students to establish the relationship between the passage of time and the variation of these mathematical functions.

Required Materials

a. Recyclable materials (cardboard, bottle caps, old CDs, etc).

b. Scissors and glue.

c. Ruler, compass, and protractor.

d. Pencils and colored pens.

e. Wall clock (as a reference).

Detailed Step-by-Step for Carrying Out the Activity

1. Gather the necessary materials.
2. The group should conduct a brief research on the sine, cosine, and tangent functions and how they relate to the dynamics of a clock.
3. Using the collected materials, the group should design and build the clock, ensuring that each function – sine, cosine, and tangent – is properly represented.
4. After construction, the functionality of the clock should be tested.
5. The group should prepare a report explaining the clock's construction process, the operation of the trigonometric functions within it, and how the activity contributed to the understanding of the content studied.

Project Deliverables and Written Report

Students must deliver the built trigonometric clock and a written report detailing the work done. The report must be at least 3 pages long and should be divided into:

1. Introduction: Description of the activity, explaining the relevance and application of trigonometry in the real world, presenting the project's objective.

2. Development: Discussion on the theory of trigonometry and its functions (sine, cosine, tangent). The construction method of the clock and how the activity was carried out should be explained in detail. The results obtained should also be presented and discussed.

3. Conclusions: Reflect on the learning achieved with the project. Which concepts were clearer with the activity? How did the project help understand the application of trigonometric functions in daily life?

4. Bibliography: Indicate all sources used for the work: books, websites, videos, among others.

Each student in the group must participate in the report's elaboration, thus ensuring a complete collaborative work. The participation of each student in the report should be clearly indicated.