Lesson Plan | Traditional Methodology | Triangles: Law of Sines

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to provide students with a clear understanding of what will be covered during the lesson. By defining the main objectives right at the beginning, students will have an overview of the skills and knowledge they should acquire. This helps direct students' attention and focus to the most important points of the lesson, promoting a more effective and structured learning.

Main Objectives

1. Understand the Law of Sines and its basic formula.

2. Apply the Law of Sines to solve problems involving any triangles.

3. Develop skills to determine measures of sides and angles in triangles using the Law of Sines.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to provide students with a clear understanding of what will be covered during the lesson. By defining the main objectives right at the beginning, students will have an overview of the skills and knowledge they should acquire. This helps direct students' attention and focus to the most important points of the lesson, promoting a more effective and structured learning.

Context

To start the lesson on the Law of Sines, explain to students that this is a fundamental mathematical tool used to solve problems involving any triangles, not just right triangles. Emphasize that the Law of Sines is particularly useful in situations where we cannot directly apply the Pythagorean Theorem or basic trigonometry. For example, in navigation, astronomy, and even some engineering fields, the correct understanding and application of this law are essential for accurately determining distances and angles.

Curiosities

Did you know that ancient navigators used principles similar to the Law of Sines to navigate the seas? Without modern technology, they relied on measurements of angles and distances to determine their location and the route to follow. This is an example of how mathematics can have fascinating practical and historical applications.

Development

Duration: (50 - 55 minutes)

The purpose of this stage is to provide students a detailed and practical understanding of the Law of Sines. By addressing the definition, derivation of the formula, and practical application, students will have the opportunity to visualize how theory translates into problem-solving. Moreover, the guided exercises and proposed questions will allow students to consolidate the knowledge acquired, applying it in varied situations.

Covered Topics

1. Definition of the Law of Sines: Explain the basic formula of the Law of Sines: a/b = sin(A)/sin(B) = sin(C)/c = 2R, where a, b, and c are the sides of the triangle, and A, B, and C are the angles opposite these sides, respectively. Detail the importance of R, which is the radius of the circumcircle of the triangle. 2. Derivation of the Formula: Show how the formula of the Law of Sines is derived using basic geometry and trigonometry. Use any triangle, draw the circumcircle, and demonstrate how the relationship is obtained. 3. Practical Application: Present practical examples of how the Law of Sines can be used to solve problems. Include an example where it is necessary to determine a side of a triangle and another where it is necessary to determine an angle. 4. Guided Exercises: Propose exercises to be solved collectively with the class. Guide students step by step in solving, highlighting important steps and common pitfalls.

Classroom Questions

1. In triangle ABC, the sides a, b, and c measure 7 cm, 10 cm, and 12 cm, respectively. Determine the value of angle A using the Law of Sines. 2. A triangle has angles A = 30° and B = 45°, and the side opposite angle A measures 10 cm. Use the Law of Sines to find the length of the side opposite angle B. 3. Triangle ABC has sides a = 8 cm and b = 6 cm, and the angle opposite side a is 60°. Calculate the angle opposite side b using the Law of Sines.

Questions Discussion

Duration: (20 - 25 minutes)

The purpose of this stage is to consolidate the knowledge gained during the lesson, allowing students to check and discuss their answers and better understand the application of the Law of Sines. The detailed discussion of solutions and students' engagement with reflective questions helps reinforce understanding and resolve any remaining doubts.

Discussion

• 📘 Discussion of the Questions

Question 1: In triangle ABC, the sides a, b, and c measure 7 cm, 10 cm, and 12 cm, respectively. Determine the value of angle A using the Law of Sines.

Solution: Apply the Law of Sines to find angle A.

First, find the sines of the angles using the formula:

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

We know that:

[ \frac{7}{\sin(A)} = \frac{10}{\sin(B)} = \frac{12}{\sin(C)} ]

To simplify, we can use the relationship between two sides and their opposite angles:

[ \frac{7}{\sin(A)} = \frac{12}{\sin(C)} ]

To find angle A, we need angle C or the relationship between the other angles and sides. Suppose we have angle C, then we can solve for A.

Question 2: A triangle has angles A = 30° and B = 45°, and the side opposite angle A measures 10 cm. Use the Law of Sines to find the length of the side opposite angle B.

Solution: Apply the Law of Sines to find the side opposite angle B.

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} ]

[ \frac{10}{\sin(30°)} = \frac{b}{\sin(45°)} ]

We know that ( \sin(30°) = 0.5 ) and ( \sin(45°) = \frac{\sqrt{2}}{2} ), so:

[ \frac{10}{0.5} = \frac{b}{\frac{\sqrt{2}}{2}} ]

[ 20 = \frac{b \cdot \sqrt{2}}{2} ]

[ b \cdot \sqrt{2} = 40 ]

[ b = \frac{40}{\sqrt{2}} ]

[ b = 20\sqrt{2} \approx 28.28 \text{ cm} ]

Question 3: Triangle ABC has sides a = 8 cm and b = 6 cm, and the angle opposite side a is 60°. Calculate the angle opposite side b using the Law of Sines.

Solution: Apply the Law of Sines to find the angle opposite side b.

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} ]

[ \frac{8}{\sin(60°)} = \frac{6}{\sin(B)} ]

We know that ( \sin(60°) = \frac{\sqrt{3}}{2} ), so:

[ \frac{8}{\frac{\sqrt{3}}{2}} = \frac{6}{\sin(B)} ]

[ \frac{16}{\sqrt{3}} = \frac{6}{\sin(B)} ]

[ \sin(B) = \frac{6\sqrt{3}}{16} ]

[ \sin(B) = \frac{3\sqrt{3}}{8} \approx 0.6495 ]

[ B \approx 40.89° ]

Student Engagement

1. ### 🗣️ Student Engagement

How can you verify if your answers using the Law of Sines are correct? What are the main difficulties you encountered when solving problems using the Law of Sines? How can the Law of Sines be applied in practical everyday situations? Give examples. If you had a triangle with all the angles known, how could you use the Law of Sines to find the sides? Discuss the importance of correctly using measurement units when applying the Law of Sines.

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to recap the main points covered during the lesson, reinforce the connection between theory and practice, and highlight the importance of the content for real situations. This moment allows students to consolidate their learning and recognize the relevance of the acquired knowledge.

Summary

• Understanding the Law of Sines and its basic formula.
• Application of the Law of Sines to problems involving any triangles.
• Solving practical problems to determine measures of sides and angles in triangles.
• Discussion on the derivation of the Law of Sines formula.
• Guided exercises to consolidate knowledge.

During the lesson, the theory of the Law of Sines was connected with practice through practical examples and guided exercises. Students were able to visualize how the formula is derived and applied in real problems, such as determining sides and angles of triangles in various mathematical and everyday situations.

Understanding the Law of Sines is crucial not only for solving mathematical problems but also for applications in navigation, astronomy, and engineering. For example, ancient navigators used similar principles to determine their routes at sea, showing how mathematical concepts can have significant impact in practice and history.