Contextualization

Trigonometry is a branch of mathematics focused on the study of the relationships between the lengths of 2 sides of a right triangle (a triangle whose angle formed by two sides is 90°). The sine, cosine, and tangent of an angle are examples of these relationships and will be the focus of this project.

Trigonometry has ancient origins and is present in various moments of human history, from the Egyptian pyramids to modern air and sea navigations. Today, this area of knowledge is fundamental in the study of physics, engineering, geography, architecture, music, computing, and many other fields.

Sine, cosine, and tangent are considered the basic trigonometric functions and are used to define the other functions. They are defined as follows: The sine (sin) of an angle in the right triangle is the ratio between the length of the opposite side and the hypotenuse. The cosine (cos) is the ratio between the adjacent side and the hypotenuse. And the tangent (tan) is the ratio between the opposite side and the adjacent side or the ratio between the sine and cosine of that angle.

Importance of Trigonometric Functions

Trigonometry is an essential tool in various applied and theoretical areas. From the daily life of a civil engineer, who needs to calculate the height of a building, to the work of an astronomer who seeks to calculate distances in the universe, trigonometric relationships are fundamental. Moreover, it is a subject constantly applied in different questions of entrance exams and competitions in Brazil.

By mastering the trigonometric relationships sin, cos, and tan, the student expands their mathematical repertoire to solve more complex problems and is able to understand physical phenomena, such as simple harmonic motion.

Practical Activity: Mathematics in Practice

Project Objective

The objective of the project is to build a sundial that is capable of marking the hours correctly, through the use of trigonometric functions - sine, cosine, tangent.

Detailed Project Description

This project will involve the discipline of Mathematics, with an emphasis on Trigonometry, and the discipline of Physics, more specifically the study of shadows and the conversion of solar time to civil time.

For this, students will use the concepts of trigonometry to calculate the correct measurements of the sundial elements, allowing this ancient instrument to mark the hours correctly. In addition, students must perform the conversion from solar time to civil time, which involves understanding the concepts of the equation of time and the apparent movement of the Sun.

Groups should be formed by 3 to 5 students and the project will have an estimated duration of more than 12 hours, including research, development, construction of the artifact, and report writing.

Necessary Materials

• Internet access for research;
• Resistant cardboard;
• Pencil, ruler, and protractor;
• Barbecue stick or any object with the same function;
• Glue and scissors.

Detailed Step-by-Step for Carrying Out the Activity

1. Research: Start by researching what a sundial is and how it works. Consult the suggested sources and take note of important information.

2. Drawing: Draw the sketch of the sundial. The sundial will be a semi-circle divided into 12 equal parts, representing the hours of the day.

3. Trigonometric Calculations: Calculate the positions of the hour lines using trigonometric functions. Apply the concepts of sine, cosine, and tangent to perform these calculations.

4. Construction: After making the calculations, build the sundial using cardboard, scissors, ruler, and other necessary materials.

5. Testing and Adjustments: Place the sundial outdoors and check its accuracy throughout the day, making necessary adjustments.

6. Report: Finally, prepare a detailed report describing the project, explaining all the steps taken, the calculations made, the challenges encountered, and how they were overcome.

Project Delivery

At the end of the work, each group must deliver:

1. The crafted sundial;

2. A written report, following the topics: Introduction, Development, Conclusions, and Bibliography.

In the Introduction, the group must contextualize Trigonometry, highlighting its applications, especially the construction of a sundial.

In Development, the group must present the calculations made for the sundial's elaboration and also how the conversion from solar time to civil time was done. Each step of the project must be described in detail.

In the Conclusions section, students must reflect on the results obtained, the functionality of the sundial they built, and the challenges encountered during the project execution.

The Bibliography is the last topic of the report, where students must indicate all the sources consulted to carry out the work.

Remembering that the evaluation goes beyond correct mathematical work, it also takes into account skills such as time management, communication, problem-solving, creative thinking, proactivity, and others.