Lesson Plan | Traditional Methodology | Basic Trigonometric Lines: 30º, 45º, 60º

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to prepare students for understanding and applying the basic trigonometric lines (sine, cosine, and tangent) at the angles of 30º, 45º, and 60º. This theoretical foundation is essential for them to solve practical problems involving right triangles, facilitating the understanding of concepts that will be explored during the lesson.

Main Objectives

1. Recognize and identify the values of sine, cosine, and tangent for the angles of 30º, 45º, and 60º.

2. Apply the values of sine, cosine, and tangent to solve problems in right triangles.

3. Understand the relationship between the angles and the sides of right triangles using trigonometric functions.

Introduction

Duration: (10 - 15 minutes)

 Purpose: The purpose of this stage is to prepare students for understanding and applying the basic trigonometric lines (sine, cosine, and tangent) at the angles of 30º, 45º, and 60º. This theoretical foundation is essential for them to solve practical problems involving right triangles, facilitating the understanding of concepts that will be explored during the lesson.

Context

 Context: To start the lesson, explain that trigonometry is one of the most important areas of mathematics, with applications in various fields such as engineering, architecture, physics, and even in computer games. It helps us understand and solve problems involving angles and distances. The angles of 30º, 45º, and 60º are particularly important and frequently appear in practical problems. Understanding the basic trigonometric lines of these angles is fundamental for success in many scientific and technological disciplines.

Curiosities

類 Curiosity: Did you know that trigonometry was essential for maritime navigation before the advent of GPS? Navigators used stars and trigonometric calculations to determine their position at sea. Moreover, today, engineers use these concepts to build bridges and skyscrapers, ensuring that structures are safe and stable.

Development

Duration: (40 - 50 minutes)

 Purpose: The purpose of this stage is to provide a detailed and applied understanding of the trigonometric functions sine, cosine, and tangent at the angles of 30º, 45º, and 60º. By the end of this stage, students should be able to apply these concepts to solve practical problems involving right triangles, solidifying the theoretical and practical foundation necessary for further study in trigonometry.

Covered Topics

1.  Specific Topics: 2. Basic Definitions of Sine, Cosine, and Tangent: Explain that sine, cosine, and tangent are trigonometric ratios that relate the angles of a right triangle to the lengths of its sides. Detail that sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. 3. Sine, Cosine, and Tangent Values for 30º, 45º, and 60º: Provide the exact values of the trigonometric ratios for the angles of 30º, 45º, and 60º. Use notable triangles (30º-60º-90º and 45º-45º-90º) to derive these values. For example, sine of 30º is 1/2, cosine of 45º is √2/2, and tangent of 60º is √3. 4. Practical Application in Right Triangles: Demonstrate how to use the values of sine, cosine, and tangent to calculate the lengths of sides in right triangles. Solve practical examples where, given an angle and one side, it is possible to find the other sides of the triangle. 5. Relationship between Angles and Sides: Explain the relationship between the angles and sides of a right triangle. Emphasize that knowing one angle besides the right angle allows determining the other elements of the triangle using trigonometric functions.

Classroom Questions

1.  Questions for Class Resolution: 2. Calculate the length of the opposite side in a right triangle where the hypotenuse is 10 units and the adjacent angle is 30º. 3. In a right triangle, one of the angles is 45º and one of the legs measures 5√2 units. Determine the length of the hypotenuse. 4. A right triangle has an angle of 60º and the adjacent leg measures 6 units. Calculate the length of the opposite side.

Questions Discussion

Duration: (20 - 25 minutes)

 Purpose: The purpose of this stage is to ensure that students fully understand the solutions to the proposed questions, clarify doubts, and promote an in-depth discussion on the practical application of trigonometric concepts. This moment is crucial for consolidating learning and motivating students to reflect on the importance and utility of trigonometric functions in real contexts.

Discussion

•  Discussion of Questions:

• Question 1: Calculate the length of the opposite side in a right triangle where the hypotenuse is 10 units and the adjacent angle is 30º.

• Steps to Solution: Using the definition of sine, we know that sine(30º) = 1/2. Thus, sine(30º) = opposite side / hypotenuse. Substituting the values, we have 1/2 = opposite side / 10. Multiplying both sides by 10, we get opposite side = 5 units.

• Question 2: In a right triangle, one of the angles is 45º and one of the legs measures 5√2 units. Determine the length of the hypotenuse.

• Steps to Solution: In a 45º-45º-90º triangle, the legs are equal and the hypotenuse is leg * √2. Therefore, if one leg measures 5√2 units, the hypotenuse will be 5√2 * √2 = 5 * 2 = 10 units.

• Question 3: A right triangle has an angle of 60º and the adjacent leg measures 6 units. Calculate the length of the opposite side.

• Steps to Solution: Using the definition of tangent, we know that tangent(60º) = √3. Thus, tangent(60º) = opposite side / adjacent side. Substituting the values, we have √3 = opposite side / 6. Multiplying both sides by 6, we obtain opposite side = 6√3 units.

Student Engagement

1. ❓ Questions and Reflections: 2. What is the importance of memorizing the values of sine, cosine, and tangent for the angles of 30º, 45º, and 60º? 3. How can understanding these trigonometric ratios help in everyday problems? 4. Discuss how you would use these concepts to solve a practical problem, such as the height of a building based on the projected shadow. 5. What was the biggest difficulty encountered in solving the questions? How can we overcome it?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to recap the main points covered during the lesson, reinforcing students' learning and ensuring they understand the importance and practical application of the concepts taught. Additionally, this stage aims to consolidate the knowledge acquired and motivate students to recognize the relevance of trigonometry in real contexts.

Summary

• Basic definitions of sine, cosine, and tangent, including their ratios and relationships with the sides of a right triangle.
• Exact values of the trigonometric ratios for the angles of 30º, 45º, and 60º, using notable triangles.
• Practical application of trigonometric functions to solve problems in right triangles.
• Relationship between the angles and the sides of a right triangle, highlighting the importance of knowing one angle and one side to determine the other elements of the triangle.

The lesson connected theory with practice by using practical examples and solved problems to demonstrate how the values of sine, cosine, and tangent can be applied to calculate the sides of right triangles. This allowed students to see the direct application of theoretical concepts in real and practical situations, reinforcing theoretical understanding with applied practice.

Understanding the basic trigonometric lines is essential for various fields of knowledge, such as engineering, architecture, and physics. In everyday life, these concepts are applicable in tasks such as measuring inaccessible heights, calculating distances, and even in navigation using geographic coordinates. The practical relevance of trigonometry makes learning these concepts fundamental for solving real problems.