Contextualization

Theoretical Introduction

The study of Trigonometry is one of the fundamental bases of Mathematics, its understanding is extremely necessary as it is used in many areas of knowledge, such as Physics, Engineering, and Architecture. Trigonometry is the study of the relationships between the angles' measures and the lengths of the sides of triangles.

The basic trigonometric lines refer to the angles of 30º, 45º, and 60º, which are commonly used in trigonometric calculations. They are special because they facilitate problem-solving by presenting exact values for the sine, cosine, and tangent functions.

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and the tangent is the ratio of the length of the opposite side to the length of the adjacent side. These functions are essential for solving problems involving right triangles.

Contextualization of the Theme

Trigonometric lines are of fundamental importance in our daily lives and in various professional areas. They are used to calculate inaccessible distances, in navigation, physics, engineering, architecture, and even in video games.

For example, engineers can use trigonometry to calculate the height of a tower or the slope of a bridge. Physicists can use it to calculate the speed of a moving object or the trajectory of a projectile. In everyday life, a simple example of application could be calculating the height of a tree or building using only a tape measure and the observation angle.

Atividade Prática

Activity Title: The Mathematics of Shadows

Project Objective

This project aims not only to review the concepts of basic trigonometric lines (30º, 45º, 60º), but also to apply them in practice to calculate the height of an object through its shadow. We will integrate Physics, through the study of Light and Shadows, and Mathematics, through trigonometry.

Detailed Project Description

The activity will be based on the creation of a series of field experiments to calculate the height of different objects (for example, a tree, a lamppost, a building) using the mathematics of shadows. You will measure the shadow of a stick of known height and the shadow of the object to be measured. With this data, using trigonometry, you will calculate the height of the object.

In addition, in this project, you also need to research the application of trigonometry in everyday practice and in some professions, such as Engineering or Architecture. This research should be included in the final report.

Necessary Materials

1. A 1-meter tall stick.
2. Tape measure.
3. Calculator.
4. Notebook for notes.
5. Digital camera or smartphone (to document the experiment).

Detailed Step-by-Step

1. Form a group of 3 to 5 students.
2. Discuss as a group and select which objects (at least 3 of different heights) will have their height calculated.
3. On a sunny day, measure the shadow of the 1-meter stick and the shadow of the chosen object.
4. Record the exact time these measurements were taken.
5. Use trigonometry to calculate the height of the object based on the shadow measurements.
6. Repeat steps 3 to 5 for the other selected objects.
7. Document the entire process with photos and notes.
8. Conduct research on the application of trigonometry in everyday practice and in some professions.
9. Write a project report following the guidelines below.

Project Report

Write a detailed project report, addressing:

1. Introduction: Provide context for the theme, its relevance, and real-world application. Describe the objective of this project.
2. Development:
   • Theory: Explain the theory of basic trigonometric lines.
   • Activity: Describe the activity in detail, including how the measurements were taken, which objects were chosen, and why.
   • Methodology: Indicate the methodology used, explaining the trigonometric formula used to calculate the heights.
   • Results: Present and discuss the results obtained. Compare with the real height of the object (if available).
   • Research: Present the result of the research conducted on the application of trigonometry.
3. Conclusion: Conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
4. Bibliography: Indicate the sources you relied on to work on the project, including books, web pages, videos, etc.

The project must be submitted within 3 weeks after its assignment.