Lesson plan of Trigonometry: Sine, Cosine, and Tangent: Review

Objectives (5 - 10 minutes)

1. Understand the concept of sine, cosine and tangent: Students should be able to clearly and concisely explain what sine, cosine and tangent are, and how these functions relate to angles in a right triangle.

2. Apply trigonometric functions to practical problems: Students should be able to solve problems involving the application of trigonometric functions, such as calculating the height of a building, the distance between two inaccessible points, among others.

3. Review and consolidate prior knowledge of trigonometry: The secondary objective is to ensure that students have a solid foundation in trigonometry, reviewing previous concepts that are fundamental to understanding the current topic.

Secondary objectives:

• Develop critical thinking and problem-solving skills: In addition to learning the concepts, students will be challenged to apply them in real-world problem situations, which will contribute to the development of their critical thinking and problem-solving skills.

• Encourage active participation and classroom collaboration: Through the proposed active methodology, students will be encouraged to actively participate in the class, sharing their questions, ideas and solutions, fostering a collaborative learning environment.

Introduction (10 - 15 minutes)

1. Review of basic concepts: The teacher should start the class by reviewing the basic concepts of trigonometry, such as what a right triangle is, the definition of the hypotenuse and the catheters. This review is crucial for students to correctly understand and apply trigonometric functions. (3 - 5 minutes)

2. Problem situations: The teacher can present two problem situations that will be solved throughout the class. The first could be determining the height of a building, where the student will have to use the tangent. The second could be determining the distance between two inaccessible points, where the student will have to use the sine and the cosine. These problem situations will serve to contextualize the importance of trigonometric functions and motivate students to learn the content. (2 - 3 minutes)

3. Contextualization: The teacher should explain the importance of trigonometry in everyday life, highlighting its applications in various areas, such as engineering, architecture, physics, geography, among others. For example, trigonometry is essential for determining the height of an inaccessible object, to design the slope of a ramp, to calculate the trajectory of a projectile, among others. (2 - 3 minutes)

4. Curiosities: The teacher can share some curiosities about the origin and development of trigonometry. For example, he can mention that the word "trigonometry" comes from the Greek "trigono" (triangle) and "metron" (measure), that is, trigonometry is the measurement of triangles. The teacher can also mention that trigonometry was developed by several ancient civilizations, such as the Egyptians, the Babylonians, the Indians and the Greeks, and that it has had practical applications since ancient times, such as in the construction of the pyramids of Egypt. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Presentation of the theory (10 - 15 minutes): The teacher should present the concepts of sine, cosine and tangent of an angle in a right triangle. It is important for him to explain how these functions are defined and how they relate to the sides of the triangle.

   • Definition of sine, cosine and tangent: The teacher should start by explaining that the sine of an acute angle in a right triangle is the ratio of the opposite leg to that angle and the hypotenuse of the triangle. The cosine is the ratio of the adjacent leg to that angle and the hypotenuse, and the tangent is the ratio of the opposite leg to the adjacent leg.
   • Relationships between functions: The teacher should explain that the sine and cosine are periodic functions, with period 2π, and that the tangent is periodic, with period π. He should also emphasize that the value of these functions depends on the angle in question, and that they vary between -1 and 1.

2. Practical demonstration (5 - 10 minutes): After presenting the theory, the teacher should make a practical demonstration to illustrate how trigonometric functions are applied. To do this, he can use a right triangle drawn on the blackboard or in a geometry software.

   • Calculating functions: The teacher should demonstrate how to calculate the sine, cosine and tangent of an angle in a right triangle. He can do this using a simple example, such as a right triangle with a 30° angle.
   • Applying the functions: The teacher should demonstrate how to apply the functions to solve practical problems. He can use the problem situations presented in the Introduction, or create new situations to illustrate different applications of trigonometric functions.

3. Fixation activity (5 - 10 minutes): After the demonstration, the teacher should propose a fixation activity so that students can practice the concepts learned.

   • Problem solving: The teacher should propose problems involving the application of trigonometric functions. He can start with simple problems, and then gradually increase the complexity of the problems.
   • Group discussion: The teacher should encourage students to discuss the solutions to the problems in groups, promoting the exchange of ideas and collaboration between students.

Feedback (10 - 15 minutes)

1. Group discussion (5 - 7 minutes): The teacher should initiate the Feedback phase by promoting a group discussion on the solutions to the problems proposed in the fixation activity. He should encourage students to share their solution strategies, the difficulties encountered and the learnings obtained. This discussion will allow the teacher to assess the students' level of understanding of trigonometric functions and identify possible gaps or misunderstandings to be addressed in future classes.

2. Connection to theory (2 - 3 minutes): After the discussion, the teacher should make the connection between the solutions to the problems and the theory presented. He should highlight how the trigonometric functions were applied in solving the problems and reinforce the most important concepts. This step is crucial for consolidating theory in practice and for reinforcing students' learning.

3. Teacher feedback (2 - 3 minutes): The teacher should then provide general feedback on the class performance. He should praise the strengths, such as active participation, collaboration and correct application of trigonometric functions, and point out areas for improvement, such as the need to practice problem solving more and to review some basic concepts. The teacher should also answer any remaining questions and clarify possible misunderstandings.

4. Final reflection (1 - 2 minutes): To conclude the class, the teacher should propose that students reflect for a minute on the following questions:

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

   This reflection will allow students to consolidate what they have learned, identify their difficulties and prepare for the next class.

5. Student feedback (1 - 2 minutes): The teacher can also ask for feedback from students about the class. He can ask what they liked the most, what they found most difficult, what they would like to learn more about and how they felt about the pace and complexity of the class. This feedback will be valuable for the teacher to adjust his pedagogical practice and improve the teaching-learning process.

Conclusion (5 - 10 minutes)

1. Summary of contents (2 - 3 minutes): The teacher should recap the main points covered during the class, reinforcing the definition of sine, cosine and tangent, their relationships to the angles in a right triangle, and how to apply them to solve practical problems. He can use the blackboard or a slide to outline and visualize these concepts, making it easier for students to understand.

2. Connection between theory and practice (1 - 2 minutes): The teacher should highlight how the class provided a deeper understanding of the theory of trigonometric functions and how these concepts were applied in practice, through the resolution of problem situations. He should emphasize the importance of understanding theory in order to be able to apply it efficiently and accurately.

3. Additional materials (1 - 2 minutes): The teacher should suggest additional study materials for students who wish to deepen their knowledge of trigonometry. These materials may include math books, educational websites, explanatory videos, among others. The teacher should emphasize that practice is fundamental to learning trigonometry, and that students should devote themselves to solving a variety of problems to consolidate their knowledge.

4. Relevance of the topic (1 - 2 minutes): Finally, the teacher should emphasize the importance of trigonometry in everyday life and in various areas of knowledge. He can mention practical examples, such as the use of trigonometry in engineering to calculate the height of a building, in architecture to design the slope of a ramp, in physics to determine the trajectory of a projectile, in geography to calculate the distance between two inaccessible points, among others. The teacher should also encourage students to recognize the presence of trigonometry in their daily lives, and to value the learning of this content as a useful and powerful tool for problem solving and understanding the world around them.