Objectives (5 - 7 minutes)

1. Understand the concept of trigonometric inequality and its importance in solving practical problems.
2. Learn to solve trigonometric inequalities using trigonometric identities and analyzing the behavior of sine and cosine functions.
3. Apply the acquired knowledge to solve practical problems involving trigonometric inequalities, such as determining the values of an angle that satisfy a given inequality.

Secondary Objectives:

• Develop logical and analytical reasoning skills in solving complex mathematical problems.
• Stimulate autonomy and perseverance in seeking solutions through practical and challenging activities.
• Reinforce previous knowledge of trigonometry, trigonometric identities, and sine and cosine functions.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher starts the lesson by briefly reviewing the concepts of trigonometry, trigonometric identities, and sine and cosine functions. This can be done through a quick review of concepts, interactive questions, or problems involving these topics. (3 - 5 minutes)

2. Problem Situations: The teacher presents two problem situations to arouse students' interest and demonstrate the relevance of the topic. The first situation could be: 'Imagine you need to find all angles that satisfy the inequality sin(x) > 0.5. How would you solve it?' The second situation could be: 'How would you determine the angles that satisfy the inequality cos(x) < -0.7?' (3 - 5 minutes)

3. Contextualization: The teacher explains the importance of trigonometric inequality in solving practical problems, such as determining angles in engineering, physics, architecture, astronomy, and other areas that use trigonometry. The teacher may also mention that the ability to solve trigonometric inequalities is fundamental knowledge for the study of mathematics and various professional fields. (2 - 3 minutes)

4. Introduction to the Topic: The teacher introduces the topic of the lesson, explaining that trigonometric inequality is an inequality that involves trigonometric functions, and the goal is to find the values of the angle that satisfy the inequality. To arouse students' curiosity, the teacher can share some curiosities or interesting applications, such as using trigonometric inequalities to model plant growth or wave motion. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity 'Inequality Path' (10 - 12 minutes):

   • Description: The teacher divides the class into groups of up to 5 students and gives each group a set of colored cards. Each card has an angle (in radians or degrees), a trigonometric inequality, and the solution to the inequality. The teacher explains that the goal of the activity is for students, in their respective groups, to organize the cards so that the inequalities and their solutions are in ascending or descending order, depending on the case.

   • Step by Step:

     1. The teacher distributes the cards and explains the rules of the activity.
     2. The groups, together, discuss and organize the cards.
     3. The teacher circulates around the room, assisting the groups when necessary and checking the progress of the activity.
     4. At the end, each group presents their sequence of cards and explains the reasoning used. The teacher makes the necessary comments and corrections, if needed.

   • Objectives: This activity aims to allow students to visualize and better understand the process of solving trigonometric inequalities. In addition, group discussion promotes interaction and the exchange of ideas among students, strengthening collaborative learning.

2. Activity 'Inequality Challenge' (10 - 12 minutes):

   • Description: The teacher presents the problem situation: 'You are a group of scientists trying to decipher an ancient manuscript containing trigonometric inequalities. If you can solve all the inequalities, you will be rewarded with the secret of the manuscript. Good luck!' The teacher then gives each group a sheet with a series of trigonometric inequalities to solve. The inequalities can vary in difficulty and may include trigonometric identities to simplify the resolution.

   • Step by Step:

     1. The groups receive the sheet with the inequalities and start solving them.
     2. The teacher circulates around the room, assisting the groups when necessary and checking the progress of the activity.
     3. At the end of the stipulated time, each group presents their solutions. The teacher makes the necessary comments and corrections, if needed.
     4. The teacher reveals the 'secret of the manuscript': a motivational message or an interesting fact about the application of trigonometric inequalities.

   • Objectives: This activity aims to provide students with the opportunity to apply what they have learned about trigonometric inequalities in a playful and challenging context. In addition, solving the inequalities promotes the development of students' critical and analytical thinking.

3. Discussion and Reflection (5 - 7 minutes):

   • Description: After the conclusion of the activities, the teacher leads a discussion in the classroom about the strategies used by the groups to solve the inequalities, the difficulties encountered, and the solutions found. The teacher also promotes reflection on the importance of trigonometric inequalities in solving practical problems and in the development of various areas of knowledge.

   • Objectives: This final discussion aims to consolidate students' learning, clarify possible doubts, and reinforce the importance of the topic covered. In addition, the exchange of ideas and reflection promotes the development of students' critical and analytical thinking.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • Description: The teacher gathers all students and asks each group to share their solutions or conclusions from the activities carried out. Each group has a maximum of 3 minutes to present. During the presentations, other students are encouraged to ask questions or make comments.

   • Objectives: This stage aims to promote the exchange of ideas and collaborative learning, allowing students to see different approaches to solving trigonometric inequalities. In addition, the practice of oral presentation helps develop communication and expression skills.

2. Connection with Theory (2 - 3 minutes):

   • Description: After all the presentations, the teacher makes the connection between the practical activities carried out and the theoretical concepts covered in the lesson. For example, the teacher can highlight how the use of trigonometric identities simplified the resolution of the inequalities, or how the analysis of the behavior of sine and cosine functions helped determine the values of the angles that satisfy the inequalities.

   • Objectives: This stage aims to consolidate students' learning, reinforcing the understanding of theoretical concepts through practice. In addition, the connection between theory and practice helps make the content more meaningful and relevant to students.

3. Final Reflection (3 - 4 minutes):

   • Description: The teacher proposes that students reflect individually on what they learned in the lesson. To do this, the teacher asks the following questions:

     1. 'What was the most important concept you learned today?'
     2. 'What questions have not been answered yet?'

   • Objectives: This stage aims to stimulate metacognition, that is, reflection on the learning process. By thinking about what they have learned and what their doubts are, students can identify gaps in their understanding and seek clarification of these doubts. In addition, the final reflection helps consolidate learning, making it more lasting and meaningful.

4. Feedback and Closure (1 - 2 minutes):

   • Description: The teacher thanks everyone for their participation and ends the lesson. The teacher can also take this opportunity to give general feedback on the lesson, highlighting strengths and areas that need more practice or study.

   • Objectives: This stage aims to reinforce the importance of feedback and proper closure for the learning process. In addition, the teacher's feedback can help students better understand what was learned and prepare for future lessons.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes):

   • Description: The teacher reviews the main points discussed during the lesson, recalling the concept of trigonometric inequality, the importance of trigonometric identities in solving inequalities, and the analysis of the behavior of sine and cosine functions to determine the values of the angles that satisfy the inequality. In addition, the teacher can also recap the strategies used during the practical activities and the difficulties encountered by students.

   • Objectives: This stage aims to consolidate students' learning, reinforcing the concepts and skills developed during the lesson.

2. Connection of Theory, Practice, and Applications (1 - 2 minutes):

   • Description: The teacher explains how the lesson connected theory (concepts of trigonometric inequality, trigonometric identities, sine and cosine functions) with practice (solving inequalities in groups, analyzing resolution strategies). The teacher also reinforces the importance of trigonometric inequalities in practical situations, such as determining angles in various areas of knowledge.

   • Objectives: This stage aims to show students the relevance of what was learned, reinforcing the applicability of the concepts and skills developed.

3. Extra Materials (1 - 2 minutes):

   • Description: The teacher suggests additional materials for students who wish to deepen their knowledge of trigonometric inequalities. These materials may include math books, educational websites, explanatory videos, and additional exercises. For example, the teacher may recommend the use of online trigonometric function simulators, which allow students to visualize the behavior of functions and test different inequalities.

   • Objectives: This stage aims to encourage autonomous and in-depth study by providing additional resources for learning.

4. Importance of the Topic (1 minute):

   • Description: To conclude the lesson, the teacher emphasizes the importance of the topic covered, mentioning again some practical applications of trigonometric inequalities and the relevance of the knowledge acquired for the study of mathematics and various professional fields.

   • Objectives: This stage aims to motivate students by showing that what they have learned has real-world applications and is relevant to their learning and their lives.