Lesson Plan | Traditional Methodology | Trigonometric Function: Graphs

Objectives

Duration: 10 - 15 minutes

The purpose of this stage of the lesson plan is to introduce students to the topic of the graphs of trigonometric functions, highlighting the main concepts and skills that will be developed throughout the lesson. This stage aims to prepare students for the understanding and practical application of trigonometric functions, facilitating their understanding of their graphs and the information that can be extracted from them.

Main Objectives

1. Describe the graphs of the trigonometric functions sine, cosine, and tangent.

2. Identify and interpret the period, amplitude, and roots of the graphs of the trigonometric functions.

3. Draw graphs of trigonometric functions based on their characteristics.

Introduction

Duration: 10 - 15 minutes

 Purpose: The purpose of this stage is to contextualize students about the importance of studying trigonometric functions and their graphs. By presenting practical applications and curiosities, this stage aims to spark students' interest and engagement, preparing them for a detailed understanding of the content that will be covered throughout the lesson.

Context

 Context: Start the lesson by explaining that trigonometric functions are fundamental in various fields, such as engineering, physics, and computer graphics. They are used to model periodic phenomena, such as sound and light waves. Tell students that by understanding the graphs of these functions, they will be able to interpret, predict, and represent real phenomena accurately and efficiently.

Curiosities

 Curiosity: Did you know that trigonometric functions are used in film animation? For example, to create realistic movements of characters and objects, animators use trigonometric functions to calculate trajectories and smooth movements, making scenes more natural and convincing. This practical application helps to understand the importance of studying these functions.

Development

Duration: 50 - 60 minutes

 Purpose: The purpose of this stage is to provide students with a detailed understanding of the graphs of trigonometric functions, highlighting their main characteristics, such as period, amplitude, and roots. By solving guided problems, students will be able to apply theoretical knowledge to practical examples, thus consolidating their understanding of the content.

Covered Topics

1.  Graph of the Sine Function: Explain that the sine function is a periodic function with a period of 2π. The graph of the sine function is a smooth wave that oscillates between -1 and 1. Highlight the main points, such as the intersections with the x-axis (the roots are multiples of π), the maximum and minimum points, and the shape of the curve. 2.  Graph of the Cosine Function: Similar to the sine function, the cosine function is also periodic with a period of 2π. The graph of the cosine function is also a smooth wave, but it starts at 1 when x = 0. Explain the main points, such as the intersections with the x-axis (the roots are multiples of π), the maximum and minimum points, and the shape of the curve. 3.  Graph of the Tangent Function: The tangent has a period of π and features vertical asymptotes where the function is undefined (multiples of π/2). The graph of the tangent function has a distinct shape and repeats every π units. Explain the intersection points with the x-axis (multiples of π), the intervals of rapid growth, and the asymptotes.

Classroom Questions

1. Draw the graph of the sine function in the interval from 0 to 2π. Identify the intersection points with the axes, the maxima, and minima. 2. Draw the graph of the cosine function in the interval from 0 to 2π. Identify the intersection points with the axes, the maxima, and minima. 3. Draw the graph of the tangent function in the interval from -π to π. Identify the intersection points with the axes and the asymptotes.

Questions Discussion

Duration: 20 - 25 minutes

 Purpose: The purpose of this stage is to review and consolidate the knowledge acquired by students throughout the lesson, ensuring that they understand the characteristics and properties of the graphs of trigonometric functions. By discussing the answers and engaging students in reflections, this stage aims to clarify remaining doubts and reinforce the practical application of the content.

Discussion

•  Discussion of the Questions:

• Graph of the Sine Function: The graph of the sine function in the interval from 0 to 2π is a smooth wave that starts at zero, reaches its maximum at π/2, returns to zero at π, reaches its minimum at 3π/2, and returns to zero at 2π. The intersections with the x-axis occur at points 0, π, and 2π. The maximum occurs at π/2 and the minimum at 3π/2.

• Graph of the Cosine Function: The graph of the cosine function in the same interval is also a smooth wave but starts at 1 when x = 0, reaches zero at π/2, reaches its minimum at π, returns to zero at 3π/2, and returns to 1 at 2π. The intersections with the x-axis occur at π/2 and 3π/2. The maximum occurs at 0 and 2π and the minimum at π.

• Graph of the Tangent Function: The tangent function in the interval from -π to π features vertical asymptotes at -π/2 and π/2, where the function is undefined. The graph crosses the x-axis at -π, 0, and π. The tangent grows rapidly in each interval between the asymptotes.

Student Engagement

1.  Student Engagement: 2. Ask students: How would you describe the main visual difference between the graphs of the sine and cosine functions? 3. Question: Why does the tangent function have vertical asymptotes and how does this affect its graph? 4. Ask students to reflect: How does changing the period affect the graph of a trigonometric function? 5. Challenge students: How could you use knowledge of the graphs of trigonometric functions to solve real-world problems, such as modeling sound waves?

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to review and consolidate the knowledge acquired by students, ensuring that they understand the characteristics and properties of the graphs of trigonometric functions. By summarizing the main points, connecting theory with practice, and highlighting the relevance of the content, this stage aims to reinforce learning and prepare students to apply knowledge in various contexts.

Summary

• Detailed explanation of the graphs of the sine, cosine, and tangent functions.
• Identification and interpretation of the period, amplitude, and roots of the graphs of trigonometric functions.
• Drawing of graphs of trigonometric functions based on their characteristics.
• Discussion about the visual differences between the graphs of the sine, cosine, and tangent functions.
• Analysis of the vertical asymptotes in the graph of the tangent function and its impact.
• Reflection on the change in the period of trigonometric functions and its effects on the graphs.

The lesson connected the theory of the graphs of trigonometric functions with practice by providing clear examples and guided problems. Students were able to apply theoretical knowledge in practical exercises, which facilitated understanding of the properties of the graphs and their usefulness in modeling real phenomena, such as sound and light waves.

The study of the graphs of trigonometric functions is crucial for various areas, including engineering, physics, and computer graphics. Understanding these graphs enables students to solve everyday problems, such as modeling periodic phenomena and creating realistic animations. For example, animators use trigonometric functions to calculate smooth and natural movements in films.