Lesson Plan | Traditional Methodology | Trigonometry: Sine, Cosine, and Tangent: Review

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to ensure that students have a clear understanding of the fundamental concepts of sine, cosine, and tangent, as well as the ability to apply them correctly in right triangles. This establishes the necessary foundation for solving more complex problems and the practical application of trigonometry.

Main Objectives

1. Reinforce the concepts of sine, cosine, and tangent.

2. Calculate the values of sine, cosine, and tangent in a right triangle.

3. Apply the concepts in a specific triangle with sides 3, 4, and 5.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to ensure that students have a clear understanding of the fundamental concepts of sine, cosine, and tangent, as well as the ability to apply them correctly in right triangles. This establishes the necessary foundation for solving more complex problems and the practical application of trigonometry.

Context

To start the lesson on trigonometry, it's essential to contextualize students about the importance and applicability of the concepts of sine, cosine, and tangent. Explain that trigonometry is a fundamental part of mathematics that studies the relationship between the angles and sides of triangles. These concepts are widely used in various fields, such as engineering, physics, architecture, and even technology, like game development and computer graphics.

Curiosities

Did you know that trigonometry has its roots in ancient astronomy? Astronomers used these concepts to calculate distances between stars and planets. Additionally, in everyday life, trigonometry is used to determine the height of buildings, the width of rivers, and even in air and sea navigation, where it's essential to calculate precise routes and distances.

Development

Duration: (55 - 60 minutes)

The purpose of this stage is to ensure that students understand and apply the concepts of sine, cosine, and tangent in right triangles through detailed explanations and practical examples. By solving problems in class, students reinforce their understanding and practice the application of the learned concepts.

Covered Topics

1. Definitions of sine, cosine, and tangent: Explain that sine, cosine, and tangent are ratios between the sides of a right triangle concerning one of its acute angles. 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. 2. Fundamental formulas: Detail the fundamental formulas for calculating sine, cosine, and tangent: sine(θ) = opposite side / hypotenuse, cosine(θ) = adjacent side / hypotenuse, tangent(θ) = opposite side / adjacent side. 3. Practical example with triangle 3, 4, and 5: Present a right triangle with sides of lengths 3, 4, and 5 (hypotenuse). Calculate the values of sine, cosine, and tangent for the acute angles of the triangle.

Classroom Questions

1. Calculate the sine, cosine, and tangent of a right triangle where the opposite side is 7, the adjacent side is 24, and the hypotenuse is 25. 2. Given a right triangle where the hypotenuse is 17 and one of the sides is 8, find the values of sine, cosine, and tangent of the angle opposite the given side. 3. In a right triangle where the opposite side to angle θ is 15 and the hypotenuse is 17, calculate the value of tangent(θ).

Questions Discussion

Duration: (15 - 20 minutes)

The purpose of this stage is to consolidate the knowledge acquired by the students, allowing them to review and discuss their answers while reflecting on the practical applicability of the concepts of sine, cosine, and tangent. This promotes a deeper understanding and engagement with the content.

Discussion

• Explain that to calculate sine, cosine, and tangent of the right triangle where the opposite side is 7, the adjacent side is 24, and the hypotenuse is 25, the following steps should be used:

• Sine(θ): Divide the opposite side by the hypotenuse. Thus, sine(θ) = 7 / 25.

• Cosine(θ): Divide the adjacent side by the hypotenuse. Thus, cosine(θ) = 24 / 25.

• Tangent(θ): Divide the opposite side by the adjacent side. Thus, tangent(θ) = 7 / 24.

• For the right triangle with hypotenuse 17 and one of the sides 8:

• Calculate the other side using the Pythagorean theorem: a² + b² = c², where a = 8 and c = 17. Thus, b = √(17² - 8²) = 15.

• Sine(θ): Divide the opposite side by the hypotenuse. Thus, sine(θ) = 15 / 17.

• Cosine(θ): Divide the adjacent side by the hypotenuse. Thus, cosine(θ) = 8 / 17.

• Tangent(θ): Divide the opposite side by the adjacent side. Thus, tangent(θ) = 15 / 8.

• For the right triangle where the opposite side to angle θ is 15 and the hypotenuse is 17:

• Calculate the adjacent side using the Pythagorean theorem. Thus, cosine(θ) = √(17² - 15²) / 17 = 8 / 17.

• Tangent(θ): Divide the opposite side by the adjacent side. Thus, tangent(θ) = 15 / 8.

Student Engagement

1. Ask the students: What is the importance of understanding sine, cosine, and tangent in real life? 2. Have the students discuss in groups practical examples where trigonometry can be used in daily life. 3. Question the students: How do you think engineers use trigonometry in the construction of bridges and buildings? 4. Propose to the students to calculate sine, cosine, and tangent of a right triangle with different sides from the given examples and discuss their answers with the class.

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to reinforce the students' learning by recapping the main points of the lesson and highlighting the connection between theory and practice. This helps to consolidate the acquired knowledge and demonstrate the relevance of the learned concepts, promoting a deeper and more lasting understanding.

Summary

• Review of the concepts of sine, cosine, and tangent.
• Definitions and fundamental formulas for sine, cosine, and tangent.
• Practical example with a right triangle of sides 3, 4, and 5.
• Resolution of practical problems involving right triangles.
• Discussion about the application of trigonometry in various practical areas.

The lesson connected theory with practice by using concrete examples of right triangles, such as the triangle with sides 3, 4, and 5, and by solving practical problems. This allowed students to visualize how the formulas for sine, cosine, and tangent are applied in real situations and reinforced the importance of trigonometry in everyday life and in various professions.

The subject presented is fundamental not only for understanding advanced mathematical concepts but also for practical applications in daily life. Curiosities such as the use of trigonometry in ancient astronomy and its relevance in modern engineering, in the construction of bridges, buildings, and even in technology, exemplify its importance and make learning more interesting and relevant for students.