Objectives (5 - 7 minutes)

1. Understand the concept of trigonometric functions and their properties, with an emphasis on the definition of inputs (angles) and outputs (values of trigonometric ratios).
2. Develop the ability to calculate and interpret trigonometric ratios in different quadrants of the trigonometric circle.
3. Practice applying trigonometric functions to real-world problems, such as calculating the height of an inaccessible object.

Secondary Objectives:

1. Stimulate students' capacity for logical and abstract reasoning through the study of trigonometric functions.
2. Promote teamwork and collaborative discussion through solving practical problems in groups.
3. Encourage autonomy and responsibility for their own learning through prior study of the content and active participation during class.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher begins the class by reviewing the concepts of sine, cosine, and tangent, and how they are calculated on the trigonometric circle. It may be useful to do a quick review of functions, as they will be the basis for the new content. The teacher can propose some quick exercises for the class to review and practice these concepts. (3 - 5 minutes)

2. Problem situations: The teacher presents two problem situations that involve the use of trigonometric functions. The first could be calculating the height of a building using the shadow it projects on the ground and the angle of incidence of the light. The second could be calculating the distance between two inaccessible points on a plot of land using a theodolite and trigonometry. The teacher asks students to think about how to solve these problems and write down their ideas. (5 - 7 minutes)

3. Contextualization: The teacher explains that trigonometric functions are used in diverse areas of knowledge, such as engineering, physics, architecture, geography, among others. He or she can give practical examples, such as the use of trigonometry to calculate the trajectory of a rocket, the height of a mountain, the slope of a roof, etc. He or she can also mention that trigonometric functions are used in various technologies, such as GPS, sonar, radar, etc. (2 - 3 minutes)

4. Introduction to the topic: The teacher presents the concept of trigonometric functions, explaining that it associates a numerical value (the trigonometric ratio) to each angle of a trigonometric circle. He or she explains that the trigonometric circle is divided into four quadrants, and that trigonometric ratios have different values in each quadrant. The teacher also mentions that trigonometric functions are periodic, that is to say, their values repeat themselves every 360º (or 2π radians). He or she can illustrate these concepts with the aid of a drawing or a physical model of the trigonometric circle. (5 - 7 minutes)

Development (20 - 25 minutes)

1. Presentation of the theory (10 - 12 minutes):

   1.1. The teacher introduces the formal definition of trigonometric functions, explaining that it is a relationship between angles and trigonometric ratios (sine, cosine, and tangent). He or she can present the sine function as an example, showing that for each angle on the trigonometric circle, there is a corresponding sine value.

   1.2. Next, the teacher explains the idea of "input" and "output" in a trigonometric function. He or she can use an analogy with a machine, where the input angle is inserted and the corresponding trigonometric ratio is the output. He or she should emphasize that, in a trigonometric function, the input is always an angle and the output is always a trigonometric ratio.

   1.3. The teacher continues explaining that trigonometric functions have a domain (set of all possible angles) and a codomain (set of all possible values for the trigonometric ratios). He or she should also mention that the image set of a trigonometric function is the set of all values that the function actually assumes.

   1.4. The teacher then moves on to explain the characteristics of trigonometric functions: that they are periodic, that they have different values in each quadrant of the trigonometric circle, and that they have maximums and minimums.

2. Problem-solving (10 - 12 minutes):

   2.1. The teacher presents the problems of the height of the building and the distance between the inaccessible points, which were proposed in the Introduction. He or she asks students to, in groups, try to solve these problems using trigonometric functions. The teacher circulates around the room, assisting the groups and answering questions.

   2.2. After a certain amount of time, the teacher gathers the class and asks one or two groups to present their solutions. He or she then shows how these problems can be solved systematically, using trigonometry. He or she should emphasize that trigonometry is a powerful tool for solving practical everyday problems.

   2.3. The teacher can also propose other problems, challenging students to apply their knowledge of trigonometric functions. For example, he or she could propose the problem of calculating the height of a balloon using the shadow it projects on the ground and the angle of incidence of the light. Or, the problem of calculating the distance between two inaccessible points at the bottom of the sea using sonar and trigonometry.

3. Discussion and Conclusion (3 - 5 minutes):

   3.1. The teacher ends the Development stage of the class by promoting a discussion about the solutions presented and the problems proposed. He or she should encourage students to share their ideas and explain how they arrived at their solutions.

   3.2. Finally, the teacher reinforces the importance of trigonometric functions and trigonometry for solving everyday problems and in various areas of knowledge. He or she can give examples of how trigonometry is used in different professions and technologies, to motivate students to continue studying and applying these concepts.

Feedback (8 - 10 minutes)

1. Summary and Recapitulation (3 - 4 minutes):

   1.1. The teacher begins this stage by doing a brief summary of the concepts and applications presented during class. He or she can review the definition of trigonometric functions, the calculation of trigonometric ratios, the application of these functions to real-world problems, and the importance of trigonometry in various areas of knowledge and technologies.

   1.2. Next, the teacher asks students to recap what they have learned, asking them to share the most important points of each topic covered. He or she should encourage students to formulate their answers in a clear and concise manner.

   1.3. The teacher can also ask direct questions to check students' understanding. For example: "What is the definition of trigonometric functions?" or "How would you solve the problem of the height of the building using trigonometry?"

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

   2.1. The teacher asks students to reflect on how the concepts learned in class connect with the real world. He or she can ask questions such as: "Can you think of other everyday situations where trigonometry could be useful?" or "How do you think trigonometry is applied in areas such as engineering, architecture, physics, etc.?"

   2.2. The teacher can also ask students to think about how the concepts learned can be applied in other contexts. For example, he or she could propose the following situation: "Imagine that you are on a ship and want to measure the distance to an island. You have a sextant to measure the angle between the horizon and the top of the island. How could you use trigonometry to calculate the distance to the island?"

3. Final Reflection (2 - 3 minutes):

   3.1. The teacher ends the class by asking students to reflect on what they have learned. He or she can ask questions such as: "What was the most important concept you learned today?" or "What questions have not yet been answered?"

   3.2. The teacher can also ask students to write down in their notebooks a question or doubt that they would like to discuss in the next class. This helps to reinforce learning and encourages students to be more active in the learning process.

   3.3. Finally, the teacher thanks the students for their participation and congratulates them on their effort and progress. He or she encourages students to continue studying and practicing, and to seek help whenever they have questions.

Conclusion (5 - 7 minutes)

1. Summary of the Content (2 - 3 minutes):

   1.1. The teacher begins the Conclusion by doing a brief summary of the main points covered during class. He or she recapitulates the definition of trigonometric functions, the relationship between angles and trigonometric ratios, the application of these functions to practical problems, and the importance of trigonometry in various areas of knowledge.

   1.2. The teacher reinforces the most important concepts, such as the definition of trigonometric functions, the interpretation of inputs and outputs, the calculation of trigonometric ratios, and problem-solving using trigonometry.

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

   2.1. The teacher highlights how the class connected theory, practice, and real-world applications. He or she mentions that, after the theoretical review, students had the opportunity to apply their knowledge to practical problems, such as calculating the height of a building or the distance between two inaccessible points.

   2.2. The teacher reinforces that trigonometry is a powerful and useful tool, with applications in various areas of knowledge and in everyday situations. He or she can mention some of these applications again, such as engineering, architecture, physics, etc.

3. Extra Materials (1 - 2 minutes):

   3.1. The teacher suggests some extra study materials for students who wish to deepen their knowledge of trigonometric functions. He or she can suggest math books, educational websites, explanatory videos, among others.

   3.2. The teacher can also suggest some additional exercises for students to practice at home. He or she should remind students that practice is fundamental for solidifying concepts and developing problem-solving skills.

4. Importance of the Subject (1 - 2 minutes):

   4.1. The teacher ends the class by reinforcing the importance of the subject for students' lives. He or she explains that, although trigonometry can seem abstract and complex, it has practical and real-world applications, which can be useful in various situations.

   4.2. The teacher can give concrete examples of how trigonometry is used in everyday life, such as in calculating the trajectories of projectiles (used in sports such as soccer, basketball, etc.), in calculating inaccessible distances (used in navigation, civil construction, etc.), or in determining inaccessible heights (used in meteorology, astronomy, etc.).

   4.3. He or she can also mention that the study of trigonometry helps to develop important skills, such as logical thinking, abstraction, problem-solving, among others. And that these skills are valuable not only for mathematics, but for life in general.