Lesson plan of Trigonometry: Double/Triple Angle

Objectives (5 - 7 minutes)

1. Understand the concept of half-angle and double-angle in trigonometry and how they relate to the Half-Angle and Double-Angle Theorems.
2. Apply the Half-Angle and Double-Angle Theorems to find the value of sine, cosine, and tangent of a half-angle or double-angle.
3. Solve practical problems that involve the Half-Angle and Double-Angle Theorems, demonstrating the ability to apply theoretical knowledge to real-world situations.

Secondary Objectives:

• Develop critical thinking and analytical skills when finding solutions to complex trigonometry problems.
• Foster teamwork and effective communication by discussing and solving problems in groups.
• Stimulate interest and curiosity in trigonometry concepts by showing their practical applicability and relevance in the real world.

The teacher should present these Objectives to students at the beginning of the class, explaining the importance of each one and what students are expected to be able to do by the end of the class.

Introduction (10 - 15 minutes)

1. Review of basic trigonometry concepts: The teacher begins the class by reviewing important trigonometry concepts, such as the trigonometric cycle, the trigonometric ratios (sine, cosine, and tangent), and their basic definitions. This review is essential for students to be able to understand and correctly apply the concepts of half-angle and double-angle.

2. Problem situations: Next, the teacher presents two situations that will serve as the basis for the Development of the topic. The first situation could be, for example, determining the value of sine, cosine, and tangent of an angle that is half or double another known angle. The second situation could be solving a practical problem involving the use of the Half-Angle or Double-Angle Theorem. These situations aim to arouse students' interest and show the topic's practical applicability.

3. Contextualization: The teacher then contextualizes the importance of trigonometry in everyday life, highlighting its applications in various fields, such as engineering, physics, architecture, navigation, among others. For example, it can be mentioned how trigonometry is used to determine the height of a building, calculate the distance between two points on a map, or predict the trajectory of a moving object.

4. Introduction to the topic: Finally, the teacher introduces the topic of half-angle and double-angle, explaining that these concepts are fundamental for solving more complex trigonometry problems and that they will be able to solve problems that previously seemed unattainable. The teacher can use visual examples, such as the movement of a hand on a clock, to illustrate the idea of an angle and how it relates to the angle. In addition, it can be mentioned that the topic of half-angle and double-angle is a natural extension of what they have already learned in trigonometry and that by mastering it, they will be one step closer to becoming masters of trigonometry.

Development (20 - 25 minutes)

1. Theory - Half-Angle and Double-Angle Concepts (8 - 10 minutes)

   • The teacher introduces the concept of half-angle and double-angle, explaining that, in a trigonometric circle, a half-angle is half of another angle and a double-angle is double another angle.

   • Next, the teacher presents the Half-Angle and Double-Angle Theorem, which states that for any angle θ, we have:

     • sin(2θ) = 2 * sin(θ) * cos(θ)
     • cos(2θ) = cos²(θ) - sin²(θ)
     • tan(2θ) = 2 * tan(θ) / (1 - tan²(θ))

   • The teacher explains each of these formulas, using examples to illustrate the application of each one.

2. Practice - Solving Exercises (10 - 15 minutes)

   • The teacher proposes a series of exercises that involve the application of the Half-Angle and Double-Angle Theorem. The exercises can vary in difficulty level, from the simplest, which involve only direct substitution in the formula, to the most complex, which require the solving of trigonometric equations.

   • Students are encouraged to solve the exercises in groups, promoting discussion and the exchange of ideas. The teacher circulates around the room, assisting groups that encounter difficulties and clarifying doubts.

   • After solving the exercises, the teacher calls attention to the key points of each exercise, reinforcing the concepts and strategies used in the solution.

3. Application - Solving Practical Problems (5 - 7 minutes)

   • The teacher proposes one or two practical problems that involve the use of the Half-Angle and Double-Angle Theorem. These problems can be, for example, determining the value of sine, cosine, and tangent of an angle that is half or double another known angle, or solving a practical problem involving the use of the Half-Angle or Double-Angle Theorem.

   • Students are encouraged to solve the problems in groups, promoting discussion and the exchange of ideas. The teacher circulates around the room, assisting groups that encounter difficulties and clarifying doubts.

   • After solving the problems, the teacher discusses the solutions with the class, highlighting the important points and the strategies used. The teacher also emphasizes the importance of verifying the reasonableness of the answers, that is, whether the values obtained make sense in the context of the problem.

Review (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher begins the Review stage by promoting a group discussion on the solutions found by students for the proposed exercises and problems. This is an opportunity for students to share their solving strategies, clarify doubts, and learn from their peers. The teacher should encourage the participation of all students, ensuring that everyone has the opportunity to speak and be heard.

   • During the discussion, the teacher should ask guiding questions that help students reflect on what they have learned, such as "What strategy did you use to solve this problem?" or "How did you know that you were on the right track to the solution?" .

2. Connection with Theory (2 - 3 minutes)

   • Next, the teacher makes the connection between the solutions found by the students and the theory presented. The teacher can, for example, ask students how they used the Half-Angle and Double-Angle Theorem to solve the problems and what difficulties were encountered. The goal is for students to realize the practical applicability of the theory and the importance of understanding fundamental concepts for solving complex problems.

3. Individual Reflection (2 - 3 minutes)

   • The teacher then proposes that students reflect individually on what they learned in class. The teacher asks provocative questions, such as "What was the most important concept you learned today?" or "What questions have yet to be answered?" .

   • Students are encouraged to write down their reflections in a notebook or sheet of paper. The teacher should emphasize the importance of reflecting on one's own learning process, identifying what was learned, which strategies were effective, and what can be improved. This reflection stage is essential for students to internalize the knowledge acquired and become autonomous learners.

4. Feedback and Closure (1 minute)

   • Finally, the teacher asks for brief feedback from students on the class. The teacher can ask, for example, "What did you think of today's class?" or "What would you like to learn more about this topic?" . This feedback is valuable for the teacher to assess the effectiveness of their teaching practice and make adjustments, if necessary, for future classes.

   • The teacher thanks the students for their participation, highlights the key points of the class, and closes the activity.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes)

   • The teacher summarizes the main points covered in the class, reinforcing the concepts of half-angle and double-angle, as well as the Half-Angle and Double-Angle Theorem.

   • The teacher may, for example, recap the formulas of the theorem, explain again how to apply them, and emphasize the importance of verifying the reasonableness of the answers.

   • The teacher can also review the practical problems solved during the class, highlighting the strategies used and the connection between theory and practice.

2. Theory-Practice Connection (1 - 2 minutes)

   • The teacher reinforces the importance of connecting theory with practice, reminding students that trigonometry is not just a set of formulas and rules, but a powerful tool for understanding and solving real-world problems.

   • The teacher may, for example, mention how the Half-Angle and Double-Angle Theorem can be used to solve problems in engineering, physics, architecture, among other areas.

3. Supplementary Materials (1 minute)

   • The teacher suggests additional study materials for students who wish to deepen their knowledge on the topic. These materials may include trigonometry books, educational websites, explanatory videos, among others.

   • The teacher may, for example, suggest that students research more about the Half-Angle and Double-Angle Theorem and try to solve other problems that involve the use of this theorem.

4. Everyday Life Application (1 - 2 minutes)

   • Finally, the teacher reinforces the relevance of the topic to students' everyday lives, recalling that trigonometry is used in various practical situations, from determining distances and heights to predicting trajectories and movements.

   • The teacher may, for example, propose that students observe and identify everyday situations in which trigonometry can be applied, such as when measuring the height of an object with a smartphone, when calculating the distance between two points on a map, or when predicting the trajectory of a ball in a game.