Objectives (5 - 7 minutes)

1. Understanding the concept of trigonometric function: The teacher must ensure that students understand what a trigonometric function is and how it relates to triangles and circles. Emphasis should be placed on the idea that trigonometric functions are used to represent the relationship between the angles of a right triangle and the ratios between the sides of that triangle.

2. Identification of periodicity: Students should be able to identify periodicity in a trigonometric function, that is, the pattern of repetition of the function's values. This involves understanding that trigonometric functions are periodic and that their values repeat at regular intervals.

3. Application of periodicity in practical problems: Students should be able to apply the concept of periodicity in practical problems. This may involve determining the values of a trigonometric function in a specific interval, identifying when a function reaches its maximum or minimum value, or predicting the behavior of a function based on its periodicity pattern.

   Secondary objectives:

   • Development of critical thinking skills: The teacher should aim to develop students' ability to analyze and interpret trigonometric functions, as well as to apply the concept of periodicity in different contexts.

   • Strengthening problem-solving skills: Through problem-solving involving trigonometric functions and periodicity, students should be encouraged to develop and enhance their problem-solving skills.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the trigonometry concepts that were previously covered, such as the definitions of sine, cosine, and tangent, and how these functions are calculated in a right triangle. This can be done through a quick review, asking questions to students and encouraging them to participate actively.

2. Initial problem situations: To arouse students' interest and demonstrate the applicability of the subject, the teacher can present two initial problem situations:

   • Problem 1: "Imagine you are looking at the clock hand pointing to the number 3. As time passes, the hand moves slowly to the right. How could you use trigonometry to determine the position of the hand at any moment?"

   • Problem 2: "Imagine you are flying in a plane and looking down at a lighthouse on the ground. The lighthouse is spinning slowly. How could you use trigonometry to determine the position of the lighthouse at any moment?"

3. Contextualization of the importance of the subject: Next, the teacher should explain how trigonometry and, more specifically, trigonometric functions, are used in various areas, from physics and engineering (for example, to model the movement of a pendulum or a planet around the sun) to medicine (for example, to model heart rate) and music (for example, to describe the shape of a sound wave).

4. Introduction of the topic and curiosities: To introduce the topic in an interesting way, the teacher can:

   • Curiosity 1: "Did you know that trigonometric functions are even used in art? For example, the famous Dutch artist M.C. Escher used trigonometric functions to create some of his artworks, such as the drawing 'Cycle' that represents the rotation of a circle."

   • Curiosity 2: "And in nature, trigonometric functions also appear. For example, the shape of many shells and snails follows a spiral called a logarithmic spiral, which is a trigonometric function."

5. Learning expectations: Finally, the teacher should set the expectations for the lesson, explaining that students will learn to identify periodicity in a trigonometric function and apply this concept in practical problems. He should emphasize that active participation and individual effort are essential for success in learning this topic.

Development (20 - 25 minutes)

1. Theory of cycles and sinusoids (5 - 7 minutes): The teacher should start this section by explaining the idea of a cycle and how it relates to trigonometric functions. This can be done through an analogy with a clock, where the hands move in complete circles, representing a cycle. Then, he should introduce the concept of sinusoids, which are functions that repeat in cycles smoothly and continuously.

   • Definition of cycles: The teacher should define cycles as a pattern of repetition that returns to the beginning after a certain interval. He should emphasize that trigonometric functions are cyclical, which means that their values repeat at regular intervals.

   • Definition of sinusoids: Next, the teacher should define sinusoids as functions that repeat in smooth and continuous cycles. He can show the graphical representation of a sine function and a cosine function, explaining that they are examples of sinusoids.

2. Introduction to the concept of period (5 - 7 minutes): After explaining the idea of cycles and sinusoids, the teacher should introduce the concept of period.

   • Definition of period: The teacher should define the period as the smallest positive value of x for which the function repeats. He can show this in the graphical representation of a sine or cosine function, highlighting the length of a cycle.

   • Relationship between cycle, period, and frequency: The teacher should explain that the period is inversely proportional to the frequency of the function, meaning that the smaller the period, the higher the frequency of the function.

3. Identification of period and frequency in trigonometric functions (5 - 7 minutes): The teacher should then teach students how to identify the period and frequency in a trigonometric function.

   • Identification of period: The teacher can show some examples of trigonometric functions and ask students to identify the period. He should emphasize that for sine and cosine functions, the period is always 2π or 360°.

   • Identification of frequency: The teacher can then ask students to identify the frequency of a function, given its graphical representation. He should remind them that for sine and cosine functions, the frequency is the inverse of the period.

4. Examples of applying the concept of periodicity (5 - 7 minutes): To reinforce the concept of periodicity and its application, the teacher should present some examples of problems involving trigonometric functions and ask students to apply what they have learned.

   • Example 1: "Given the function f(x) = sin(x), determine the value of x for which f(x) = 1." The teacher can show students how the identification of the period and frequency can help them answer this question.

   • Example 2: "Given the function g(x) = cosine(x), determine the value of x for which g(x) = 0." The teacher can ask students to solve this problem on their own, encouraging them to apply what they have learned about the periodicity of trigonometric functions.

Throughout the Development, the teacher should encourage active student participation by asking questions, requesting examples, and encouraging them to discuss the material. He should also provide feedback and guidance as needed, ensuring that students understand the material and are able to apply it effectively.

Return (10 - 15 minutes)

1. Review of concepts (5 - 7 minutes): The teacher should start this stage by reviewing the main concepts that were covered during the lesson. This can be done interactively, asking students to review the definitions of trigonometric function, cycle, sinusoids, period, and frequency. He can ask them to give examples of each of these concepts and explain how they relate.

2. Connection between theory and practice (3 - 5 minutes): Next, the teacher should help students make the connection between the theory that was presented and the practice. He can do this by asking questions such as:

   • "How does the concept of cycle and sinusoids apply to the trigonometric functions we use to calculate the ratios between the sides of a right triangle?"

   • "How does the concept of period help us determine the value of a trigonometric function at a specific point?"

   • "How does the concept of frequency help us predict the behavior of a trigonometric function?"

   The teacher should encourage students to answer these questions in their own words, discussing among themselves and with the teacher. He should also provide feedback and guidance as needed, ensuring that students understand the connection between theory and practice.

3. Verification of understanding (2 - 3 minutes): The teacher should then check students' understanding of the concepts presented. He can do this by asking questions such as:

   • "What is the definition of a cycle?"

   • "How can you determine the period of a trigonometric function from its graphical representation?"

   • "How does the concept of frequency relate to the period of a function?"

   The teacher should ask students to answer these questions individually, encouraging them to think about the answers and express them in their own words.

4. Reflection on learning (2 - 3 minutes): Finally, the teacher should ask students to reflect on what they have learned during the lesson. He can do this by asking questions such as:

   • "What was the most important concept you learned today?"

   • "What questions have not been answered for you yet? What would you like to learn more about on this topic?"

   The teacher should give students a minute to think about these questions and then ask some of them to share their answers with the class. He should encourage students to be honest and open in their responses, reminding them that reflecting on learning is an important part of the learning process.

Throughout the Return, the teacher should emphasize the importance of individual effort, active participation, and critical thinking for success in learning mathematics and, more specifically, trigonometry. He should also provide feedback and guidance as needed, ensuring that students understand the material and are able to apply it effectively.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes): The teacher should start the Conclusion of the lesson by summarizing the main points that were covered. He can highlight the definition of trigonometric function, the concept of cycle and how it relates to sinusoidal functions, the idea of period and frequency, and the application of periodicity in practical problems. The teacher can do this through a quick review, recalling the main points and clarifying any final doubts students may have.

2. Connection between theory, practice, and applications (1 - 2 minutes): Next, the teacher should emphasize the connection between the theory that was presented, the practice that was done, and the applications of the concept of periodicity in real-world situations. For example, he can comment on how the ability to identify the period and frequency in a trigonometric function can be useful for predicting the behavior of cyclical phenomena in nature, technology, or other areas.

3. Additional materials (1 - 2 minutes): The teacher should then suggest some additional study materials for students who wish to deepen their knowledge of trigonometric functions and periodicity. This may include mathematics and trigonometry books, educational websites with tutorials and interactive exercises, and explanatory videos on YouTube or other online learning platforms.

4. Importance of the subject (1 minute): Finally, the teacher should summarize the importance of the subject for everyday life and other areas of study. For example, he can comment on how trigonometry and, more specifically, trigonometric functions, are used in various professions and fields of study, from engineering and physics to music and medicine. The teacher should emphasize that although trigonometry may seem abstract and complex, it has real and practical applications that can be useful and relevant to students in their daily lives and future careers.