Homework 7 Law Of Sines Answers

Homework 7: Law of Sines Answers – A Comprehensive Guide

Many students find trigonometry challenging, and the Law of Sines is often a major hurdle. This comprehensive guide provides detailed solutions and explanations for a hypothetical Homework 7 assignment focusing on the Law of Sines. We'll cover various problem types, focusing on clarity and understanding, helping you master this important concept. Remember, understanding the why behind the solutions is crucial for long-term retention.

Understanding the Law of Sines

Before we dive into the problems, let's refresh our understanding of the Law of Sines. This fundamental law of trigonometry connects the angles and sides of any triangle (whether it's right-angled or not). The formula is:

a/sin A = b/sin B = c/sin C

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite to sides a, b, and c respectively.

This law is incredibly useful for solving triangles when you know:

• Two angles and one side (AAS or ASA): This is often called the "angle-angle-side" (AAS) or "angle-side-angle" (ASA) case.
• Two sides and an angle opposite one of them (SSA): This is the ambiguous case, where you might have zero, one, or two possible solutions. We'll explore this in detail below.

Homework 7 Problems and Solutions

Let's tackle some sample problems that might appear in a typical Homework 7 assignment on the Law of Sines. Remember, these are examples; your specific problems might vary slightly.

Problem 1: AAS Case

Problem: In triangle ABC, angle A = 40°, angle B = 60°, and side a = 8 cm. Find the lengths of sides b and c.

Solution:

1. Find angle C: Since the sum of angles in a triangle is 180°, C = 180° - 40° - 60° = 80°.

2. Use the Law of Sines: We can use the ratios involving sides 'a' and 'b', and 'a' and 'c' respectively:

   • a/sin A = b/sin B => 8/sin 40° = b/sin 60°
   • a/sin A = c/sin C => 8/sin 40° = c/sin 80°

3. Solve for b: b = (8 * sin 60°) / sin 40° ≈ 10.79 cm

4. Solve for c: c = (8 * sin 80°) / sin 40° ≈ 12.22 cm

Problem 2: ASA Case

Problem: A triangle has angles A = 35° and B = 70°, and the side between them (c) measures 12 meters. Find the lengths of sides a and b.

Solution:

1. Find angle C: C = 180° - 35° - 70° = 75°.

2. Use the Law of Sines:

   • c/sin C = a/sin A => 12/sin 75° = a/sin 35°
   • c/sin C = b/sin B => 12/sin 75° = b/sin 70°

3. Solve for a: a = (12 * sin 35°) / sin 75° ≈ 7.16 meters

4. Solve for b: b = (12 * sin 70°) / sin 75° ≈ 11.76 meters

Problem 3: The Ambiguous Case (SSA)

Problem: In triangle ABC, a = 10, b = 12, and A = 45°. Solve the triangle (find all possible solutions).

Solution: This is the ambiguous case (SSA), meaning there might be zero, one, or two possible triangles.

1. Use the Law of Sines to find angle B:

   • a/sin A = b/sin B => 10/sin 45° = 12/sin B

   • sin B = (12 * sin 45°) / 10 ≈ 0.8485

2. Find possible values for angle B: Using a calculator (or trigonometric tables), we find two possible values for B:

   • B₁ ≈ 58.0° (using the inverse sine function)
   • B₂ ≈ 180° - 58.0° = 122.0° (since sine is positive in both the first and second quadrants)

3. Check for feasibility:

   • If B₁ ≈ 58.0°: C₁ = 180° - 45° - 58.0° = 77.0°. Then, using the Law of Sines, we can find c₁.

   • If B₂ ≈ 122.0°: C₂ = 180° - 45° - 122.0° = 13.0°. Then, using the Law of Sines, we can find c₂.

   Important Note: In the SSA case, if the calculated angle B is greater than 180° - A, there is only one solution. If the calculated angle B is less than 180° - A, there are two possible solutions (as shown in this case).

4. Solve for c₁ and c₂ using the Law of Sines:

   • c₁ = (a * sin C₁) / sin A ≈ 14.62
   • c₂ = (a * sin C₂) / sin A ≈ 3.28

Therefore, there are two possible triangles that satisfy the given conditions.

Advanced Applications and Considerations

The Law of Sines is not just a theoretical concept; it finds practical application in various fields:

• Surveying: Determining distances and angles in land surveying.
• Navigation: Calculating distances and bearings in maritime or aviation navigation.
• Engineering: Solving geometrical problems in structural engineering and construction.
• Astronomy: Calculating distances and positions of celestial bodies.

Common Mistakes to Avoid:

• Unit Consistency: Ensure all units are consistent throughout the problem (e.g., all lengths in meters, all angles in degrees).
• Rounding Errors: Avoid premature rounding during calculations. Round only your final answers to the appropriate number of significant figures.
• Ambiguous Case: Pay close attention to the SSA case, as it often leads to multiple solutions. Always check the feasibility of your solutions.
• Calculator Settings: Make sure your calculator is set to the correct angle mode (degrees or radians).

Further Practice and Resources

This guide provides a solid foundation for understanding and applying the Law of Sines. To truly master this concept, consistent practice is key. You should attempt more practice problems covering various scenarios, including the ambiguous case (SSA). There are many excellent online resources and textbooks that offer additional problems and explanations. Remember, understanding the underlying principles, not just memorizing formulas, will greatly improve your problem-solving abilities.

This extended guide aims to be a complete resource for tackling your homework assignments related to the Law of Sines. Remember to focus on understanding the processes involved in each step, not just the final numerical answers. Consistent practice and careful attention to detail will lead to mastery of this crucial trigonometric concept. Good luck!