Contextualization

The Trigonometric Inequality is a relevant topic in many fields of mathematics and its applications are vast, ranging from the analysis of sound waves to quantum physics. The study of trigonometric inequalities allows us to understand the behavior of many natural and technological phenomena, such as predicting tides and the performance of electronic circuits. Regarding the content, this project will address concepts of inequalities, trigonometric functions, solving inequalities containing sines, cosines, and tangents, and the relationship between inequalities and function graphs.

The introduction to these concepts is important for several reasons. First, mathematics, besides being present in everyday life, is used to describe and predict behaviors of physical, biological, economic phenomena, among others. That is why these concepts are frequently used in the fields of engineering, physics, computer science, economics, and many others. In addition, solving trigonometric inequalities develops critical thinking skills and problem-solving abilities.

Introduction

Inequalities are mathematical statements that assert that two quantities are not equal, in other words, they express that one value is greater or less than another. In the case of trigonometric inequalities, they involve expressions containing trigonometric functions (sine, cosine, tangent, etc.), which are used to represent and solve problems involving angles and triangles.

Trigonometric functions are used to describe wave behavior, therefore, they are fundamental tools in physics and engineering to analyze phenomena such as light and sound. Moreover, they are of great importance for the understanding and description of rotation and circular motion, fundamental elements in the study of dynamics in physics.

Trigonometric expressions can become more complex when involved with inequalities. Solving a trigonometric inequality means finding all angles that satisfy the inequality. Such resolutions require a solid understanding of trigonometric functions and their properties.

Practical Activity

Activity Title: Solving Trigonometric Inequalities in the Real World

Project Objective

The objective of the project is to apply the concepts of trigonometric inequalities in solving problems involving real-world scenarios. Students should apply their mathematical skills, particularly regarding trigonometric functions and inequalities, while developing teamwork, communication, time management, and critical thinking skills.

Detailed Project Description

In this project, groups of 3 to 5 students should choose and model a real-world problem that can be described and solved using trigonometric inequalities. Some examples of real-world problems may include identifying when during the day the shadow of a building exceeds a certain length, determining the angles of light incidence on a solar panel to optimize energy generation, or analyzing tidal heights at different times.

It is important that this project be carried out within a timeframe of at least 2 weeks, considering the complexity of the topic, the need for research, and group work.

Required Materials

• Mathematics books or online resources for research and study.
• Graphing software (such as Geogebra or Desmos) to visualize trigonometric functions and their inequalities.
• Materials for project documentation, such as paper, pens, computer, etc.

Step by Step

1. Group Formation: Divide into groups of at least 3 and at most 5 students. Each group should choose a leader who will be responsible for group organization and communication with the teacher.

2. Choosing the Real-World Problem: Each group should research and choose a real-world problem that can be modeled by trigonometric inequalities. After choosing the problem, the group should submit it to the teacher for approval.

3. Study and Research: After the problem is approved, the groups should study the relevant trigonometric functions and inequalities to solve the problem. Students can use textbooks, online classes, and other available resources.

4. Modeling and Solving: Once the functions and inequalities are understood, students should model and solve the problem. They should graph the functions and explain how the solutions of the inequalities apply to the problem.

5. Documentation: Each group should produce a detailed project report, which will include: introduction (problem contextualization, relevance, and project objective), development (theory, activity, methodology, results), and conclusion (learnings and conclusions). The groups should also compile a bibliography of the resources used.

6. Presentation: Finally, each group will present their project to the class, explaining the chosen problem, how they modeled and solved the problem, the results obtained, and the conclusions.

Project Deliverables

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

1. A written report detailing all project steps, from choosing the problem to applying trigonometric inequalities to solve the problem, including all phases mentioned in the step-by-step.

2. Oral presentation of the project to the class. During this presentation, group members are expected to demonstrate not only mastery of trigonometry content but also the skills developed during the project, such as teamwork, communication, and problem-solving.

3. Graphs and solutions of trigonometric inequalities. They must be clearly identified and explained within the report and presentation.

4. Feedback from the group on the project, including challenges encountered and how they were overcome, and what was learned in the process. This feedback can be included in the conclusion of the report.

Remember, the goal of this project is not only to solve trigonometric inequalities but also to develop important socio-emotional skills, such as teamwork, time management, critical thinking, and communication.