Objectives (5 - 7 minutes)

1. Understand the concept of Trigonometric Equation: The goal is for students to understand what a trigonometric equation is, how it is formed, and what it looks like in its general form. They should be able to identify and differentiate a trigonometric equation from other equations.

2. Learn to solve Trigonometric Equations: Students should be able to solve simple and complex trigonometric equations using the properties of trigonometric functions and trigonometric identities. They should be able to identify the primary solutions and general solutions of an equation.

3. Apply knowledge in problem-solving situations: Students should be able to apply the acquired knowledge in solving practical problems and real-world contexts involving trigonometric equations. They should be able to translate a problem into a trigonometric equation and then solve the equation to find the solution.

   Secondary Objectives:

   • Develop critical thinking skills: Solving trigonometric equations requires logical and analytical thinking. Therefore, students will have the opportunity to develop their critical thinking skills during the lesson.

   • Promote teamwork: The inverted classroom is a great opportunity for students to work in teams, discussing and solving problems together. This will help promote collaboration and communication among students.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher starts the lesson by reviewing previous concepts that are fundamental for understanding the current topic. Thus, the teacher will review the concepts of trigonometric functions, trigonometric identities, and the use of the unit circle. This can be done through brief review activities, such as solving simple equations involving trigonometric functions or identifying trigonometric identities in expressions.

2. Problem Situations: Next, the teacher will present two problem situations to instigate the curiosity and attention of the students. The first situation could be solving a physics problem that involves determining an angle from a trigonometric equation. The second situation could be a daily life problem that involves determining an inaccessible height through a shadow and the use of trigonometric functions.

3. Subject Contextualization: The teacher will explain the importance of trigonometric equations, showing how they are widely used in various areas of knowledge, such as mathematics, physics, engineering, architecture, etc. This can be done through practical examples, such as determining distances, heights, angles, and modeling periodic phenomena.

4. Introduction to the Topic: To capture the students' attention, the teacher can present some curiosities and interesting applications of trigonometric equations. For example, it can be mentioned how the ancient Egyptians used trigonometry to build the pyramids, or how trigonometry is used in predicting tides and maritime navigation. Additionally, the teacher can present the most famous equation in mathematics, Euler's Identity, which combines the five main mathematical constants (0, 1, π, e, and i) into a single equation.

   • Curiosity 1: "Did you know that trigonometry was originally developed for the study of triangles? It was only generalized for the study of periodic functions in the 18th century!"
   • Curiosity 2: "Did you know that trigonometry is widely used in medicine? It is used, for example, to model the electrical activity of the heart (electrocardiogram) and the brain (electroencephalogram)!"

Development (20 - 25 minutes)

1. Interactive Unit Circle: The teacher should divide the class into groups of up to 5 students. Each group will receive a large unit circle, made of cardboard or cardstock. The circle should be divided into 360 degrees, and at each angle, there should be a small piece of Velcro. The teacher will provide small fabric triangles with Velcro at each vertex, representing the notable angles (30, 45, 60, 90, 180, 270, 360). Students should place the triangles on the corresponding angles on the circle, forming a kind of giant "puzzle." This exercise will help students visualize the relationship between angles and trigonometric functions and understand how angles repeat on the circle.

   • Step by step:
     1. The teacher gives each group a unit circle and the small fabric triangles.
     2. Students should discuss in their groups which triangle corresponds to each angle and place the triangles on the circle.
     3. The teacher circulates around the room, observing and guiding the groups as necessary.
     4. When all groups finish, the teacher asks a representative from each group to explain the arrangement of the triangles on the circle and the relationship with trigonometric functions.

2. Group Problem Solving: The teacher provides each group with a series of problems involving trigonometric equations. The problems can be contextualized, such as determining an angle in a physics or engineering problem, or they can be abstract, focusing on the application of trigonometric identities. Students should work together to solve the problems, using the interactive unit circle and trigonometric identities as tools.

   • Step by step:
     1. The teacher gives each group a sheet with the problems and a blank sheet for notes.
     2. Students should read and discuss the problems in their groups, identifying the strategies that will be used to solve them.
     3. Students start solving the problems, noting their calculations and discussions on the blank sheet.
     4. The teacher circulates around the room, observing and guiding the groups as necessary.
     5. When all groups finish, the teacher asks a representative from each group to explain the solution to one of the problems.

3. Role-playing Activity: To make the lesson more playful, the teacher can propose a role-playing activity, where students take on the role of "trigonometric equations." Each student receives a sheet with a trigonometric equation written on it, and they must "move" around the room, looking for other students whose equations can be solved together. For example, if a student has the equation sin(x) = cos(x), they must find another student with the equation tan(x) = 1. Students should work together to solve the equations, discussing and justifying their strategies. This activity will reinforce students' ability to identify and solve trigonometric equations, and also promote collaboration and communication among students.

   • Step by step:
     1. The teacher gives each student a sheet with a trigonometric equation written on it.
     2. The teacher explains the activity and the rules: students must "move" around the room looking for other students whose equations can be solved together, and they must work together to solve the equations.
     3. Students start "moving" around the room, looking for other students.
     4. When two students with compatible equations meet, they sit together and start solving the equations, discussing and justifying their strategies.
     5. When all pairs finish, the teacher asks some pairs to explain how they solved their equations.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should gather all students and open a group discussion. Each group will have up to 3 minutes to share their solutions or conclusions from the activities. This is an opportunity for students to learn from each other, as they may be exposed to different approaches and strategies for solving trigonometric equations. The teacher should encourage students to explain their answers and justify their solutions, promoting communication and collaboration among students.

   • Step by step:
     1. The teacher gathers all students in a circle or in a format that allows everyone to see and hear each other.
     2. The teacher asks each group to share their solutions or conclusions from the activities.
     3. While the groups are sharing, the teacher should ask questions to encourage explanation and justification of the solutions.
     4. The teacher should ensure that all students have the opportunity to speak and that all groups have the opportunity to share.

2. Connection with Theory (2 - 3 minutes): After the group discussion, the teacher should quickly review the theoretical concepts covered in the lesson, explaining how they apply to the problems solved by the students. The teacher should highlight the resolution strategies used by the students that are directly related to the theoretical concepts, reinforcing the connection between theory and practice.

   • Step by step:
     1. The teacher quickly reviews the theoretical concepts, highlighting the resolution strategies used by the students that are directly related to the concepts.
     2. The teacher explains how the theoretical concepts apply to the problems solved by the students, reinforcing the connection between theory and practice.
     3. The teacher asks questions to check students' understanding and clarify any remaining doubts.

3. Final Reflection (3 - 4 minutes): To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. The teacher can ask guiding questions for reflection, such as "What was the most important concept you learned today?" and "What questions have not been answered yet?" Students should write down their answers, which can be shared with the class or kept for the next lesson. This reflection activity will help students consolidate what they have learned and identify areas of difficulty that need to be addressed in future lessons.

   • Step by step:
     1. The teacher proposes that students reflect individually on what they have learned.
     2. The teacher asks guiding questions for reflection, such as "What was the most important concept you learned today?" and "What questions have not been answered yet?".
     3. Students write down their answers.
     4. The teacher may ask some students to share their answers with the class, if there is time and if students feel comfortable doing so.
     5. The teacher ends the lesson, thanking the students for their participation and reminding them of tasks or readings for the next lesson.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes): The teacher should recap the main points covered during the lesson, recalling the definition of trigonometric equation, its practical applications, the strategies to solve them, and how these strategies were applied in the group activities. The teacher can use a whiteboard or slide presentation to highlight the most important concepts and key formulas that students should remember.

   • Step by step:
     1. The teacher reviews the main concepts, such as the definition of trigonometric equation, its properties, and the strategies for resolution.
     2. The teacher highlights the key formulas and trigonometric identities used during the lesson.
     3. The teacher reinforces the main points from the group discussions and practical activities, highlighting the skills and knowledge that students developed during the lesson.

2. Theory-Practice Connection (1 - 2 minutes): The teacher should emphasize how the lesson connected theory with practice. This can be done by recalling the group problem-solving activities and the role-playing activity, which allowed students to apply theoretical concepts in practical situations. The teacher should highlight how the understanding of theory helped students solve the problems and how the practical application of concepts helped consolidate learning.

   • Step by step:
     1. The teacher explains how the lesson connected theory with practice, recalling the activities carried out and how they allowed students to apply theoretical concepts.
     2. The teacher highlights how the practical application of concepts helped consolidate learning and understand the relevance of theory.

3. Extra Materials (1 - 2 minutes): The teacher should suggest additional study materials for students who wish to deepen their understanding of trigonometric equations. These materials may include books, websites, videos, and interactive apps that offer detailed explanations, solved examples, and additional exercises. For example, the teacher may suggest an explanatory video on solving trigonometric equations, an app that allows students to explore the unit circle interactively, or a math book that contains a section dedicated to trigonometric equations.

   • Step by step:
     1. The teacher suggests some additional study materials, briefly explaining what each material offers.
     2. The teacher encourages students to explore these materials in their own time to reinforce what they learned in the lesson and clarify any remaining doubts.

4. Everyday Application (1 minute): Finally, the teacher should emphasize the importance of trigonometric equations in everyday life, mentioning some practical applications. This can help students realize the relevance of what they have learned and motivate them to continue studying the subject.

   • Step by step:
     1. The teacher briefly mentions some practical applications of trigonometric equations, explaining how they are useful in real-life situations.
     2. The teacher concludes the lesson, thanking the students for their participation and encouraging them to continue studying and practicing.