Objectives (5 - 7 minutes)

1. The students will be able to understand and explain the Law of Cosines, which states that the square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle.
2. The students will be able to apply the Law of Cosines to solve problems involving non-right triangles, including finding the lengths of sides and measures of angles.
3. The students will be able to use the Law of Cosines to determine whether a triangle is acute, obtuse, or right-angled.

Secondary Objectives:

1. The students will develop their problem-solving skills by applying the Law of Cosines to real-world scenarios.
2. The students will enhance their understanding of trigonometric functions, specifically cosine, and its relationship with the Law of Cosines.
3. The students will improve their mathematical communication skills by explaining their reasoning and solutions to the class.

Introduction (8 - 10 minutes)

1. To begin the lesson, the teacher will remind the students of the previous trigonometry lessons, specifically the definitions of sine, cosine, and tangent, and the use of these functions in right triangles. The teacher will ask a few quick review questions to ensure the students have retained this knowledge.

2. The teacher will then present two problem situations to the class. The first problem could be about a scenario where a person is tracking a boat in the sea using a radar that can only detect the distance and angle of the boat from the radar station. The second problem could be about a triangle with known sides and one known angle, and the task is to find the missing side and the other angles. These problems will serve as starters for the development of the Law of Cosines.

3. The teacher will contextualize the importance of the Law of Cosines by explaining its applications in various fields such as physics, engineering, and architecture. For instance, in physics, the Law of Cosines is used in the study of forces in a non-right triangle. In architecture, it can be used to calculate the lengths of diagonal beams in a roof.

4. The teacher will introduce the topic of the Law of Cosines with two interesting facts:

   • The Law of Cosines is a generalization of Pythagoras' theorem to non-right triangles.
   • The Law of Cosines is useful for solving triangles when we do not know the measurement of a right angle.

5. The teacher will then draw the students' attention to the triangle on the board and ask the students to think about how they would find the length of the side c (the side opposite the angle C) if they only know the lengths of sides a and b, and the measure of the angle C.

6. To further engage the students, the teacher will share a real-world application of the Law of Cosines: In navigation, the Law of Cosines can be used to determine the distance between two points (given their coordinates) on the earth's surface. This is important in GPS systems and in calculating the shortest distance between two points on the globe.

7. The teacher will conclude the introduction by stating that by the end of the lesson, the students will be able to solve such problems using the Law of Cosines.

Development (20 - 25 minutes)

1. The Law of Cosines (8 - 10 minutes):

   1. The teacher will start by introducing the Law of Cosines, stating that it is a formula used to find the lengths of the sides of a non-right triangle when the lengths of two sides and the measure of the included angle are known.

   2. The formula will be written on the board and explained in detail: c² = a² + b² - 2ab * cos(C)

   3. The teacher will then explain the meaning of each term in the Law of Cosines.

      • c represents the unknown side opposite to the known angle C.
      • a and b represent the lengths of the other two sides of the triangle.
      • C represents the measure of the included angle between sides a and b.

   4. The teacher will emphasize that the Law of Cosines is a generalization of the Pythagorean theorem, which is used to find the lengths of the sides of a right triangle.

2. Deriving the Law of Cosines (5 - 7 minutes):

   1. The teacher will derive the Law of Cosines using the concept of vectors.

   2. The teacher will draw a triangle and explain that the displacement vectors from one vertex to the other two vertices can be added to get the displacement vector from one vertex to the other.

   3. By using the dot product of the vectors, the teacher will show how the Law of Cosines can be derived.

   4. The teacher will provide a short proof of the formula for the Law of Cosines, which will help the students understand the origins and logic behind the formula.

3. Applying the Law of Cosines (7 - 8 minutes):

   1. The teacher will then proceed to explain how to use the Law of Cosines to find the unknown side or the measure of an angle in a non-right triangle.

   2. The teacher will demonstrate solving a problem step by step on the board, using the formula and explaining each step and calculation in detail.

   3. The teacher will explain that if they know the lengths of the two sides and the measure of the included angle, they can plug these values into the Law of Cosines to find the length of the third side.

   4. The teacher will also demonstrate how the Law of Cosines can be used to find the measure of an angle when the lengths of the three sides are known.

4. Additional Properties of the Law of Cosines (3 - 5 minutes):

   1. The teacher will then explain that the Law of Cosines can also be used to classify triangles as acute, right, or obtuse, based on the values obtained from the formula.

   2. The teacher will clarify that if the square of the longest side (c) is greater than the sum of the squares of the other two sides (a and b), the triangle is obtuse.

   3. If the square of the longest side (c) is less than the sum of the squares of the other two sides (a and b), the triangle is acute.

   4. If the square of the longest side (c) is equal to the sum of the squares of the other two sides (a and b), the triangle is right.

Feedback (5 - 7 minutes)

1. Reflection:

   1. The teacher will start by asking the students to reflect on what they have learned during the lesson. They will be asked to think about how the Law of Cosines connects to their previous knowledge of right triangles and trigonometric functions.

   2. The teacher will then ask the students to consider the real-world applications of the Law of Cosines that were discussed during the lesson. They will be encouraged to think about other scenarios where they might need to calculate the lengths of sides or measures of angles in non-right triangles.

   3. The teacher will also prompt the students to reflect on the process of deriving the Law of Cosines. They will be asked to think about how understanding the derivation of a formula can help them better understand and remember the formula itself.

2. Assessment:

   1. The teacher will ask the students to take a moment to answer a few questions to assess their understanding of the Law of Cosines. These questions could include a problem to solve using the Law of Cosines, a multiple-choice question about the properties of triangles, or a short answer question about the derivation of the Law of Cosines.

   2. The teacher will then collect and review the students' responses. This will provide immediate feedback on the students' understanding and allow the teacher to address any misconceptions or areas of confusion.

3. Connection to Everyday Life:

   1. To wrap up the lesson, the teacher will ask the students to consider how the Law of Cosines can be applied in their everyday lives. For example, they may use it in activities such as video gaming (in game development, for calculating distances and angles), sports (in tracking the trajectory of a ball), or in art (in creating geometric designs).

   2. The teacher will also explain that understanding the Law of Cosines can help them appreciate the complexity of the world around them, from the design of buildings and bridges to the calculations behind modern technology like GPS.

4. Final Thoughts:

   1. The teacher will conclude the lesson by reminding the students that the Law of Cosines is a powerful tool in trigonometry that allows them to solve problems in a wide range of fields.

   2. The teacher will encourage the students to continue practicing the Law of Cosines on their own and to come to class prepared with any questions or areas they would like to review in the next lesson.

   3. The teacher will also remind the students that they can always reach out for help outside of class if they are struggling with the material, reinforcing a supportive and collaborative learning environment.

Conclusion (5 - 7 minutes)

1. Recap (1 - 2 minutes):

   • The teacher will summarize the main points of the lesson, including the Law of Cosines formula, its derivation, and its application in solving problems related to non-right triangles.
   • The teacher will also recap the additional properties of the Law of Cosines, such as its use in classifying triangles as acute, obtuse, or right-angled.

2. Connection of Theory, Practice, and Applications (2 - 3 minutes):

   • The teacher will explain how the lesson connected theory (the derivation and formula of the Law of Cosines), practice (the problem-solving activities during the lesson), and applications (the real-world scenarios where the Law of Cosines can be applied).
   • The teacher will emphasize that understanding the theory behind the Law of Cosines helps in its practical application, and that the ability to use the Law of Cosines in real-world scenarios demonstrates a deep understanding of the concept.

3. Additional Materials (1 - 2 minutes):

   • To further reinforce the students' understanding of the Law of Cosines, the teacher will recommend some additional resources. These could include online tutorials, interactive trigonometry tools, and practice problem sets.
   • The teacher will also suggest that the students review their textbook's chapter on trigonometry, specifically the section on the Law of Cosines, for additional practice and clarification.

4. Importance of the Law of Cosines (1 - 2 minutes):

   • The teacher will conclude the lesson by emphasizing the importance of the Law of Cosines in everyday life and various fields of study.
   • The teacher will remind the students that the Law of Cosines is not just a mathematical concept, but a tool that can be used in many practical applications, such as navigation, physics, engineering, and architecture.
   • The teacher will also highlight that learning the Law of Cosines helps in developing problem-solving skills, logical thinking, and mathematical reasoning, which are valuable skills for their academic and professional future.
   • The teacher will encourage the students to continue exploring and applying the Law of Cosines, and to be aware of its presence in their surroundings, further underlining the importance and relevance of the topic.