Introduction

Trigonometry is a branch of mathematics that studies the relationships between angles and measurements of a triangle. This includes trigonometric functions such as sine, cosine, and tangent. Understanding these functions is essential for solving complex problems in various fields of science, such as Physics, Engineering, and even Biology. Through trigonometric functions, we can establish relationships and solve equations and inequalities.

Trigonometric inequalities, specifically, deal with the relationship of 'greater than' or 'less than' between trigonometric expressions. Just like first-degree inequalities you studied in algebra, solving trigonometric inequalities involves finding the set of values that make the expression true. However, instead of dealing only with real numbers, we are now dealing with trigonometric functions.

Even though we may find this subject somewhat abstract, trigonometric inequalities are very useful in various practical situations. For example, they can be used to model and solve problems involving periodic movements, such as the swinging of a pendulum, the rotation of the Earth, or ocean tides.

Contextualization

In our daily lives, we use trigonometry and, consequently, trigonometric inequalities at various moments without even realizing it. For example, when we are watching a soccer game and the goalkeeper throws the ball at a specific angle to reach the maximum possible distance, or when an engineer calculates the maximum height a building can have to not cast a shadow larger than allowed on another building.

Furthermore, trigonometric inequalities are also used in many areas of science and technology. In physics, for example, they are used to describe waves, from sound to light. In engineering, they are applied in control systems, telecommunications, signal processing, among others.

The following resources are reliable and can be used to delve deeper into the topic:

1. Book: 'Plana and Spherical Trigonometry' by José Ruy Giovanni and José Roberto Bonjorno.
2. Website: Khan Academy - Trigonometry
3. Video: 'Trigonometric Inequalities' on YouTube

Practical Activity

Activity Title: Rotation and Shadow - Practical Application of Trigonometric Inequalities

Project Objective

The objective of this project is to apply trigonometric inequalities in solving a practical problem. Students should use their knowledge of trigonometry to investigate a real situation where sine, cosine, and tangent are essential. The activity should be carried out in groups of 3 to 5 students and will last 2 to 4 hours per participant.

Detailed Project Description

Students will investigate how the height of a building and the position of the Sun throughout the day affect the shadow cast by that building. The challenge will be to determine the maximum height a building can have so that its shadow does not exceed a certain distance at a specific time of the day. To do this, they should use trigonometric inequalities.

Required Materials

1. Notebook for notes and calculations.
2. Pencil and eraser.
3. Calculator.
4. Computer with internet access for research and report writing.
5. Graph paper or graphing software (optional).

Detailed Step-by-Step for Activity Execution

1. Problem Definition: Decide as a group the maximum distance the shadow can reach and the time of day when this restriction must be respected. For example, the shadow cannot exceed 100 meters at 4 p.m. Record this information.

2. Research: Find out what the angle of elevation of the Sun is in relation to the horizon at the chosen time in your city.

3. Calculation and Resolution: Use the tangent of the angle found in the research stage and the maximum distance the shadow can reach to calculate the maximum height of the building. Remember, the tangent of an angle in a right triangle is the ratio between the opposite side (height of the building) and the adjacent side (length of the shadow).

4. Presentation of Results: Clearly document all calculations performed and the conclusion reached.

5. Report: Each group should prepare a report with a clear exposition of the problem, detailing the steps taken, presenting the calculations and results, and drawing conclusions.

Deliverables and Connection with Suggested Activities

At the end of the project, each group should submit a report containing the following topics:

1. Introduction: circumstances of the problem, relevance and real-world application, and the objective of this project.

2. Development: detailed explanation of the theory used (trigonometric inequalities), the activity in detail, the methodology for solving the problem (calculations performed and the step-by-step of how they were done), and the presentation and discussion of the results obtained.

3. Conclusion: summary of the main points, the learnings obtained, and the conclusions drawn from the project.

4. Bibliography: the sources used in the project, such as books, web pages, videos, etc.

Additionally, students should give a brief oral presentation (10 min) of the project results. In this presentation, they should explain the problem, show the project steps and the results obtained. The report should be concise, objective, and clear.