Objectives (5 - 10 minutes)

1. Promote the understanding of the concept of Trigonometry, highlighting its practical application in daily life, especially in situations involving angle and distance measurements.

2. Develop the ability to identify and use trigonometric relationships (sine, cosine, and tangent) in problems involving the Right Triangle, focusing on determining measurements of sides and angles.

3. Strengthen students' ability to solve complex mathematical problems, encouraging logical reasoning and the application of learned formulas and concepts.

Secondary objectives:

• Stimulate collaboration among students, encouraging the discussion of ideas and the joint resolution of problems.
• Foster students' autonomy, promoting the use of individual learning strategies.
• Evaluate students' progress at the end of the class in order to identify possible gaps in understanding and plan future reinforcement activities.

Introduction (10 - 15 minutes)

1. Review of previous contents: The teacher should start the class by reviewing the concepts of triangles, especially the right triangle, and the definition of sine, cosine, and tangent. This review can be done through a quick questionnaire, involving students in the discussion and resolution of proposed problems. (5 minutes)

2. Problem situation 1: "Imagine you are in a building and need to measure its height. However, you do not have a ladder tall enough to reach the top. How could you use trigonometry to solve this problem?" This question is intended to arouse students' interest, showing that trigonometry is not just an abstract mathematical tool, but can be used in everyday situations. (3 minutes)

3. Contextualization: The teacher should then explain that trigonometry is used in various areas of everyday life, such as architecture, engineering, navigation, physics, among others. For example, in architecture, architects use trigonometry to calculate the height of a building or the slope of a roof. In engineering, it is used in the design of bridges and tall buildings. In navigation, trigonometry is used to determine the position of a ship or airplane. (2 minutes)

4. Problem situation 2: "Imagine you are on a boat in the middle of the ocean, with no visible reference point. How could you use trigonometry to determine your position?" This question aims to show students the importance of trigonometry in navigation, a very relevant practical application. (3 minutes)

5. Presentation of the topic: Finally, the teacher should introduce the topic of the lesson - Trigonometry in the Right Triangle. Explain that in this type of triangle, the measurements of the sides are directly related to the internal angles, and that these relationships are called trigonometric relationships. (2 minutes)

Development (20 - 25 minutes)

1. Activity "Around the World": The teacher should divide the class into groups of 4 to 5 students. Each group will receive a world map and a set of information that includes the latitude, longitude, and distance between two cities. The challenge is to determine the actual distance between the two cities on the map, using trigonometric relationships in the right triangle. (10 - 15 minutes)

   • The teacher should guide the students to choose a pair of cities on the map and identify the latitude and longitude of those cities.
   • Then, the students must calculate the difference in latitude and longitude, which will be used as angle measurements in the formed right triangles.
   • Using trigonometric relationships, the students must calculate the actual distance between the two cities.
   • This process should be repeated for several pairs of cities, allowing students to compare actual distances with distances on the map.

2. Activity "Measuring the School's Height": In this activity, students will be challenged to determine the height of the school building using trigonometry. The teacher should take the students outside the classroom, where they can see the building in its entirety. Each group of students will receive a protractor, a tape measure, and a clinometer (or a clinometer app on their phone). (5 - 10 minutes)

   • Students must measure the horizontal distance from the building to their location, the height from the protractor to eye level, and the angle formed between the horizontal and the line of sight to the top of the building.
   • After collecting these measurements, students must use trigonometric relationships to calculate the height of the building.
   • The results of each group should be compared and discussed in the classroom, allowing students to see how the accuracy of measurements and calculations affects the result.

3. Activity "Solving Everyday Problems": To conclude the practical part of the class, the teacher should propose a series of everyday problems that can be solved using trigonometry in the right triangle. Students, still in groups, should discuss and solve these problems, applying trigonometric relationships. Examples of problems may include determining the height of a streetlight pole, the depth of a well, the slope of a roof, among others. (5 - 10 minutes)

   • The teacher should circulate around the room, assisting groups that encounter difficulties.
   • At the end of the activity, each group should present the solution to at least one problem to the class, explaining the reasoning used and the calculations performed.
   • The teacher should take advantage of these presentations to reinforce the concepts and applications of trigonometry in the right triangle.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher should gather all students for a group discussion. Each group will have up to 3 minutes to share their solutions or conclusions from the activities carried out. During the presentation, the teacher should encourage other students to ask questions and express their opinions. This will promote the exchange of ideas and debate, strengthening the understanding of the topic.

2. Connection to Theory (2 - 3 minutes): After the presentations, the teacher should briefly review the trigonometric relationships and how they were applied in the activities. It is important to emphasize how theory was used in practice, reinforcing the relevance of the learned content.

3. Individual Reflection (2 - 3 minutes): The teacher should propose that students reflect individually on the class. To do this, questions such as:

   1. What was the most important concept learned today?
   2. What questions have not been answered yet?
   3. How can you apply what you learned today in real situations?
   4. What were the main difficulties encountered during the activities?
   5. Which problem-solving strategies were most effective?

4. Sharing Reflections (1 - 2 minutes): After individual reflection, the teacher should invite some students to share their answers with the class. This will allow the teacher to have immediate feedback on what the students learned and which questions still need to be clarified.

5. Closing the Class (1 minute): To conclude the class, the teacher should summarize the main points discussed and thank everyone for their participation and effort. The teacher should also remind students about the importance of reviewing the content at home and clarifying any doubts that may arise.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion by recapping the main points covered during the class. This includes the definition of trigonometry, the review of trigonometric relationships (sine, cosine, and tangent), and the application of these relationships in the right triangle. The teacher should reinforce that trigonometry is a powerful tool for solving practical problems involving angle and distance measurements.

2. Connection between Theory and Practice (1 - 2 minutes): Next, the teacher should highlight how the class managed to connect theory with practice. The teacher should emphasize that the activities carried out allowed students to apply the theoretical concepts of trigonometry in solving real problems, such as calculating the height of a building or the distance between two cities on the map. The teacher should emphasize that the ability to apply theory in practice is essential for developing mathematical skills and understanding the relevance of mathematics in various areas of knowledge.

3. Additional Materials (1 - 2 minutes): The teacher should then suggest additional study materials for students to deepen their knowledge on the topic. These materials may include explanatory videos, math learning websites, textbooks, among others. The teacher should encourage students to explore these materials on their own, emphasizing that effective learning requires constant practice and review.

4. Importance of the Topic (1 - 2 minutes): Finally, the teacher should discuss the importance of the topic covered for students' daily lives. The teacher should reinforce that trigonometry is an essential tool in various professions and everyday activities, such as architecture, engineering, navigation, physics, among others. The teacher should also remind students that trigonometry, like mathematics in general, develops valuable skills, such as logical reasoning, problem-solving ability, and accuracy in calculations. The teacher should end the class by emphasizing that, although trigonometry may seem difficult at first, with practice and effort, students will be able to master the subject and apply it in various life situations.