Objectives (5 minutes)

1. Understand the concept of Trigonometric Ratios: The teacher must ensure that students have a clear understanding of what trigonometric ratios (sine, cosine, and tangent) are and how they relate to the angles of a right triangle.

2. Apply trigonometric ratios in practical problems: Through examples and exercises, students should be able to apply trigonometric ratios to solve practical problems, such as determining the measurement of a side or an angle in a right triangle.

3. Understand the importance of trigonometric ratios: The teacher should contextualize the importance of trigonometric ratios, explaining how they are applied in various areas of science and engineering, such as navigation, physics, and architecture.

Secondary Objectives:

• Develop critical thinking and problem-solving skills: Through problem-solving involving trigonometric ratios, students should be encouraged to develop their critical thinking and problem-solving skills.

• Promote active student participation: The teacher should encourage active student participation during the lesson by asking questions, stimulating discussions, and conducting practical activities.

Introduction (10 - 15 minutes)

1. Review of relevant content (5 minutes): The teacher should start the lesson by briefly reviewing the following concepts that are fundamental to the study of trigonometric ratios: right triangles, Pythagorean theorem, and the definition of sine, cosine, and tangent. This can be done through a quick classroom discussion or by using an interactive board to demonstrate the concepts.

2. Problem-solving situations (5 minutes): Next, the teacher should present two problem-solving situations that will be addressed throughout the lesson. These situations should involve the use of trigonometric ratios to solve real problems. For example, a problem-solving situation could be: 'A building is 20 meters high. How far from the building should you be for the angle of inclination of the top of the building to be 30 degrees?' Or: 'A plane is flying at an altitude of 10,000 feet. If the pilot looks down and sees a lake forming a 45-degree angle with the horizon, what is the distance from the plane to the lake?'

3. Contextualization (2 - 3 minutes): The teacher should then explain how trigonometric ratios are used in practice, citing examples from areas such as navigation, engineering, architecture, and physics. For example, he can mention how navigators use trigonometric ratios to determine the position of a ship at sea, or how engineers use these ratios to design bridges and tall buildings.

4. Attention gain (2 - 3 minutes): To spark students' interest, the teacher can share some curiosities or interesting applications of trigonometric ratios. For example, he can mention how the ancient Egyptians used trigonometry to build the pyramids, or how trigonometry is used nowadays to design special effects in movies and video games. Additionally, he can also show how trigonometry is used in various professions, from airplane pilots to meteorologists.

Development (20 - 25 minutes)

1. Practical Activity with Ropes and Triangles (10 - 12 minutes):

   • Necessary materials: Ropes of different lengths and balloons.
   • The teacher should divide the class into groups of up to 5 students and give each group a rope of different length.
   • Next, the teacher should instruct the students to form a right triangle with the rope and a balloon, so that the balloon is the vertex of the right angle.
   • Each group should measure the sides of the triangle and the angle formed by the balloon using a protractor.
   • After measuring the values, the students should calculate the trigonometric ratios (sine, cosine, and tangent) for the angle formed by the balloon.
   • Finally, the groups should compare the results and discuss why the trigonometric ratios are constant for the same angle, regardless of the triangle's size.

2. Research and Presentation Activity (10 - 12 minutes):

   • Necessary materials: Internet access or reference books.
   • The teacher should divide the class into groups of up to 5 students and assign each group a topic related to trigonometric ratios (for example, 'Applications of Trigonometric Ratios in Engineering', 'History of Trigonometry', 'Other Trigonometric Functions').
   • Each group should research the topic and prepare a brief presentation for the class, which should include at least one practical example of the application of trigonometric ratios.
   • After the research and preparation, each group should present their topic to the class. During the presentations, the teacher should encourage the participation of all students by asking questions to verify their understanding of the subject.

3. Resolution of the Problem-Solving Situations Presented in the Introduction (5 - 7 minutes):

   • The teacher should revisit the problem-solving situations presented at the beginning of the lesson and, together with the students, apply the trigonometric ratios to solve them.
   • During the resolution of the problem-solving situations, the teacher should explain step by step the reasoning used, ensuring that the students understand the process of applying the trigonometric ratios.
   • After solving the problem-solving situations, the teacher should review the main concepts covered in the lesson and clarify any doubts that the students may have.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • The teacher should gather all students and promote a group discussion about the solutions found by each team for the practical activities and problem-solving situations.
   • During the discussion, the teacher should highlight the different approaches used and encourage students to explain the reasoning behind their solutions.
   • The teacher should reinforce the importance of communication and teamwork in solving mathematical and scientific problems, and how this applies to the use of trigonometric ratios.

2. Connection with Theory (3 - 5 minutes):

   • After the group discussion, the teacher should make the connection between the practical activities carried out and the theory presented in the lesson.
   • The teacher can highlight how the measurement of the sides and angles of the right triangle and the application of trigonometric ratios are concrete examples of how theory can be applied to solve practical problems.
   • The teacher should emphasize that understanding the theory is essential for the correct application of trigonometric ratios and for problem-solving.

3. Individual Reflection (2 - 3 minutes):

   • To conclude the lesson, the teacher should propose that students make an individual reflection on what they have learned.
   • The teacher can ask questions such as: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • Students should have a minute to think about the questions and then those who feel comfortable can share their answers with the class.
   • The goal of this activity is to have students consolidate what they have learned and identify any gaps in their understanding, which can be addressed in future lessons.

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

   • To conclude the lesson, the teacher should thank the students for their participation and effort, and reinforce the importance of continuous study and practice for learning trigonometric ratios.
   • The teacher can also ask for quick feedback from the students about the lesson, asking if they found the content and activities interesting and useful, and if they had any specific difficulties they would like to see clarified in future lessons.

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 lesson. He should emphasize the concept of trigonometric ratios (sine, cosine, and tangent), how they apply to a right triangle, and how they can be used to solve practical problems. Additionally, he should review the practical activities carried out and how they contributed to the understanding of theoretical concepts.

2. Connection between Theory, Practice, and Applications (2 - 3 minutes): Next, the teacher should emphasize how the lesson connected the mathematical theory of trigonometric ratios with the practice of measuring the sides and angles of a right triangle. He should highlight how the application of these concepts can be useful in various real-life situations, such as in engineering, architecture, physics, and navigation.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest additional materials for students to deepen their knowledge of trigonometric ratios. These materials may include textbooks, explanatory videos online, math websites, and interactive exercise programs. The teacher should encourage students to explore these resources on their own and to use the time between lessons to review the content and practice the concepts learned.

4. Importance of the Subject (1 - 2 minutes): Finally, the teacher should reinforce the importance of trigonometric ratios in daily life and in various professions. He can cite concrete examples of how these ratios are used in real situations, such as in the construction of buildings, maritime and aerial navigation, weather forecasting, among others. Additionally, the teacher can emphasize how mastering trigonometric ratios can be useful for students not only in mathematics but also in other disciplines and in their daily lives.