Lesson Plan | Technical Methodology | Trigonometric Lines of 75º, 15º, 22.5º

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to ensure that students understand the importance of specific trigonometric functions and how to apply them in practical contexts. By connecting these calculations to real-life situations, students develop skills that are directly applicable in the job market, especially in areas that require mathematical precision and critical analysis, such as engineering, architecture, and information technology.

Main Objectives

1. Understand and calculate the trigonometric functions of angles 75º, 15º, and 22.5º.

2. Apply these calculations to practical problems and real-life situations.

Side Objectives

1. Develop skills in solving complex mathematical problems.
2. Foster the ability to work in teams and share knowledge.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to ensure that students understand the importance of specific trigonometric functions and how to apply them in practical contexts. By connecting these calculations to real-life situations, students develop skills that are directly applicable in the job market, especially in areas that require mathematical precision and critical analysis, such as engineering, architecture, and information technology.

Contextualization

Trigonometric functions are fundamental to various areas of knowledge and practical application. For example, in civil construction, angles are essential to ensure the stability and safety of structures. Furthermore, in technology and graphic design, precise angle calculations are used to create realistic graphics and simulations. Understanding how to calculate the trigonometric functions of specific angles, such as 75º, 15º, and 22.5º, is a valuable skill applicable in many professional contexts.

Curiosities and Market Connection

Did you know that trigonometric functions are used to calculate the distance between stars? Astronomers use these calculations to determine the exact position of celestial bodies. In the job market, civil engineers use these functions to design and build bridges and buildings with precise angles, ensuring safety and efficiency. Architects also rely on these skills to create innovative and functional designs.

Initial Activity

Design a quick challenge! Ask students: 'How would you solve the problem of calculating the height of a building using only an angle of 15º and the distance from the base of the building?' Encourage a brief group discussion and ask them to share their ideas.

Development

Duration: 55 - 60 minutes

The purpose of this stage is to provide a practical and applied understanding of the trigonometric functions of angles 75º, 15º, and 22.5º. By engaging students in practical activities and challenges, they can consolidate their knowledge and develop skills that are directly applicable in the job market. Additionally, solving fixation exercises will help assess each student's individual understanding.

Covered Topics

1. Trigonometric functions of 75º
2. Trigonometric functions of 15º
3. Trigonometric functions of 22.5º
4. Practical applications of trigonometric functions
5. Applied problem solving

Reflections on the Theme

Guide students to reflect on how knowledge of trigonometric functions can be applied in everyday situations and the job market. Encourage them to think of areas like engineering, architecture, and technology, where the precision of angle calculations is crucial. Ask: 'How can understanding the trigonometric functions of specific angles facilitate solving complex problems in these areas?'

Mini Challenge

Practical Challenge: Building a Sloped Roof

Students will be divided into groups and must calculate the dimensions of a sloped roof using the trigonometric functions of angles 75º, 15º, and 22.5º. They will need to determine the height of the roof, the length of the beams, and the correct slope.

Instructions

1. Divide the class into groups of 4 to 5 students.
2. Distribute materials such as graph paper, rulers, calculators, and spreadsheets.
3. Each group must choose an angle (75º, 15º, or 22.5º) to work on.
4. Groups should calculate the height of the roof and the length of the beams using the trigonometric functions.
5. Guide the groups to draw the roof design on graph paper.
6. Ask each group to present their project and explain the calculations performed.

Objective: Apply knowledge of trigonometric functions in a practical project, developing skills in calculation, teamwork, and communication.

Duration: 35 - 40 minutes

Evaluation Exercises

1. Calculate the sine, cosine, and tangent of angles 75º, 15º, and 22.5º.
2. Solve the problem: An engineer needs to calculate the height of a tower using an angle of 15º and a distance of 50 meters from the base of the tower. What is the height of the tower?
3. Determine the length of the shadow of a 20-meter-high building when the angle of elevation of the sun is 22.5º.
4. Calculate the distance between two points on a Cartesian plane, knowing that the line connecting them forms an angle of 75º with the x-axis and the horizontal projection is 30 meters.

Conclusion

Duration: 15 - 20 minutes

The purpose of this stage is to consolidate the knowledge acquired during the class, promoting critical reflection on the practical application of trigonometric functions. By summarizing the main contents and discussing their implications, students reinforce their understanding and perceive the relevance of the topic for their academic lives and future careers.

Discussion

Promote an open discussion with students about the class, encouraging them to share their reflections on the topic, the practical challenge, and the exercises performed. Ask how they think knowledge of the trigonometric functions of specific angles can be applied in their future careers or in everyday situations. Encourage them to think of concrete examples and discuss how the skills developed in the class can be useful in different professional contexts.

Summary

Summarize the main contents presented in the class, including the definition and calculation of the trigonometric functions of 75º, 15º, and 22.5º, and their practical applications. Reinforce the importance of understanding these concepts to solve complex problems in various fields such as engineering, architecture, and technology. Recap the steps of the practical challenge and the fixation exercises, highlighting the key points and methods employed.

Closing

Explain how the class connected theory, practice, and applications, demonstrating the relevance of trigonometric calculations in real contexts. Reinforce the importance of the topic in everyday life, emphasizing its applications in engineering projects, architectural constructions, and other areas that require mathematical precision. Conclude the class by highlighting the importance of continuing to practice these concepts to develop critical and applicable skills in the job market.