Navigating Bearings: Mastering Law of Sines and Cosines in Real-World Problems

PreAlgebra Grades High School 7:36 Video
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Lesson Description

This lesson plan uses a video to guide students through solving bearing word problems using the Law of Sines and Law of Cosines. Students will learn to interpret bearing directions, draw accurate diagrams, and apply trigonometric principles to find distances and angles.

Video Resource

Bearing Word Problem Find Distance and Angle Using Law of Sines and Law of Cosines

Mario's Math Tutoring

Duration: 7:36
Watch on YouTube

Key Concepts

  • Bearing: Understanding and interpreting different notations for bearing (e.g., degrees clockwise from North, North/South X degrees East/West).
  • Law of Sines: Applying the Law of Sines to solve for unknown sides and angles in non-right triangles.
  • Law of Cosines: Applying the Law of Cosines to solve for unknown sides and angles in non-right triangles when given side-angle-side (SAS) or side-side-side (SSS) information.
  • Alternate Interior Angles: Recognizing and utilizing alternate interior angles formed by parallel lines and transversals to find angle measures.
  • Diagrammatic Representation: Accurately drawing and interpreting diagrams representing bearing problems.

Learning Objectives

  • Students will be able to accurately interpret and represent bearing directions in a diagram.
  • Students will be able to apply the Law of Sines and Law of Cosines to solve for unknown distances and angles in bearing problems.
  • Students will be able to convert between different notations for expressing bearing.
  • Students will be able to identify and utilize geometric relationships (e.g., alternate interior angles) to find missing angle measures.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definitions of bearing and the Law of Sines and Law of Cosines. Briefly discuss real-world applications of navigation and surveying.
  • Video Viewing (15 mins)
    Watch the Mario's Math Tutoring video on Bearing Word Problems. Encourage students to take notes on the problem-solving process, especially the diagram construction and application of the Law of Sines and Law of Cosines.
  • Guided Practice (20 mins)
    Work through the example problem from the video step-by-step, emphasizing the following: 1. Drawing the Diagram: Stress the importance of accurate diagrams, showing North, East, South, and West, and the correct angles for the bearings. 2. Finding Missing Angles: Guide students in using geometric principles (alternate interior angles, supplementary angles) to find missing angles within the diagram. 3. Applying the Law of Cosines: Demonstrate how to identify when the Law of Cosines is appropriate (SAS or SSS) and correctly apply the formula. 4. Applying the Law of Sines: Demonstrate how to identify when the Law of Sines is appropriate (AAS or ASA) and correctly apply the formula. 5. Calculating Bearing: Explain how to determine the final bearing from City C to City A, considering both clockwise from North and North/South X degrees East/West notations.
  • Independent Practice (15 mins)
    Provide students with similar bearing word problems to solve independently. Encourage them to use the same problem-solving process outlined in the video and guided practice. Provide assistance as needed.
  • Wrap-up and Assessment (5 mins)
    Review the key concepts of the lesson and answer any remaining questions. Administer the multiple-choice and fill-in-the-blank quizzes to assess student understanding.

Interactive Exercises

  • Bearing Diagram Challenge
    Provide students with a list of bearings (e.g., 30 degrees, North 45 degrees East, South 60 degrees West) and have them draw corresponding diagrams. Then, have them swap diagrams with a partner and check for accuracy.
  • Law of Sines vs. Law of Cosines Sorting Activity
    Provide students with a set of triangle problems (some requiring the Law of Sines, others the Law of Cosines). Have them sort the problems into the appropriate category based on the given information.

Discussion Questions

  • Why is it important to draw an accurate diagram when solving bearing problems?
  • In what situations is the Law of Cosines more suitable than the Law of Sines, and vice versa?
  • How can understanding geometric principles (e.g., alternate interior angles) simplify bearing problems?
  • What are some real-world applications of solving bearing problems?

Skills Developed

  • Problem-solving: Applying trigonometric concepts to solve real-world problems.
  • Spatial Reasoning: Visualizing and interpreting spatial relationships in diagrams.
  • Analytical Skills: Breaking down complex problems into smaller, manageable steps.
  • Critical Thinking: Selecting the appropriate trigonometric law based on the given information.

Multiple Choice Questions

Question 1:

Bearing is typically measured as an angle clockwise from which direction?

Correct Answer: North

Question 2:

When solving a triangle with Side-Angle-Side (SAS) information, which law is most appropriate?

Correct Answer: Law of Cosines

Question 3:

Which of the following is NOT a valid way to express a bearing?

Correct Answer: West 20 degrees North

Question 4:

If two parallel lines are cut by a transversal, what is the relationship between the alternate interior angles?

Correct Answer: They are congruent.

Question 5:

In a bearing problem, what is the first step you should typically take to solve the problem?

Correct Answer: Draw a diagram

Question 6:

Which law can be used to solve for missing sides and angles of oblique (non-right) triangles?

Correct Answer: Law of Cosines

Question 7:

When is the Law of Sines applicable to a triangle?

Correct Answer: When two angles and a non-included side are known

Question 8:

What does 'bearing' refer to in navigation problems?

Correct Answer: The direction of travel

Question 9:

If an angle measures 50 degrees clockwise from North, what is its bearing angle?

Correct Answer: 50 degrees

Question 10:

Which of the following is a key feature to include when sketching a bearing problem?

Correct Answer: The North, East, South, West directions

Fill in the Blank Questions

Question 1:

The Law of ______ is used when you know two sides and the included angle of a triangle.

Correct Answer: Cosines

Question 2:

When expressing bearing as 'North X degrees West', X represents the angle rotated towards the ______ from North.

Correct Answer: West

Question 3:

Angles that lie on opposite sides of a transversal intersecting two parallel lines are called ______ interior angles.

Correct Answer: alternate

Question 4:

The Law of ______ is used when you have angle-side pairs in a triangle.

Correct Answer: Sines

Question 5:

In navigation, ______ refers to the direction of travel measured from a reference point.

Correct Answer: bearing

Question 6:

If an angle measures 220 degrees clockwise from North, it's in the ______ quadrant.

Correct Answer: third

Question 7:

A(n) ______ diagram helps to represent a bearing problem correctly.

Correct Answer: accurate

Question 8:

An alternative way to define bearing is by stating it as ____ or south and an angle.

Correct Answer: north

Question 9:

When all three sides are known, the law of ______ is used to determine missing angles.

Correct Answer: cosines

Question 10:

The bearing from A to B is 45 degrees. The bearing from B to A is ______ degrees.

Correct Answer: 225

Educational Standards

PC.T.2.1:[Objective] Create models for situations involving trigonometry. PC.T.2.2:[Objective] Apply the Law of Sines and Law of Cosines to solve problems. PC.F.1.1:[Objective] Interpret characteristics of a function defined by an expression in the context of the situation.

Teaching Materials

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