Law Of Sines Example Problems

Mastering the Law of Sines: Example Problems and Comprehensive Guide

The Law of Sines is a crucial tool in trigonometry, enabling us to solve for unknown sides and angles in any triangle, not just right-angled triangles. Understanding and applying this law is fundamental for various fields, including surveying, navigation, and engineering. This comprehensive guide will delve into the Law of Sines, providing detailed explanations, step-by-step solutions to example problems, and addressing frequently asked questions. By the end, you'll confidently tackle any Law of Sines problem that comes your way.

Understanding the Law of Sines

The Law of Sines states the relationship between the sides and angles of any triangle:

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

Where:

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

This formula holds true for all triangles, regardless of whether they are acute, obtuse, or right-angled. The beauty of the Law of Sines lies in its versatility; it allows us to solve for missing parts of a triangle given sufficient information. We typically need at least one side and its opposite angle, along with one other piece of information (another side or angle).

There are essentially two main types of problems you'll encounter when using the Law of Sines:

1. ASA (Angle-Side-Angle): You know two angles and the included side.
2. AAS (Angle-Angle-Side): You know two angles and a side that is not included between the angles.

Let's explore several example problems, demonstrating the application of the Law of Sines in different scenarios.

Example Problem 1: ASA (Angle-Side-Angle)

Problem: A triangle has angles A = 40°, B = 60°, and the side between them, c = 10 cm. Find the lengths of sides a and b.

Solution:

1. Identify the knowns: We have A = 40°, B = 60°, and c = 10 cm.

2. Find the third angle: Since the sum of angles in a triangle is 180°, we can find angle C: C = 180° - A - B = 180° - 40° - 60° = 80°

3. Apply the Law of Sines: We'll use the ratios involving the known side (c) and its opposite angle (C):

   a/sin A = c/sin C

   a/sin 40° = 10/sin 80°

   a = (10 * sin 40°) / sin 80°

   a ≈ 6.53 cm

   Similarly, for side b:

   b/sin B = c/sin C

   b/sin 60° = 10/sin 80°

   b = (10 * sin 60°) / sin 80°

   b ≈ 8.79 cm

Therefore, a ≈ 6.53 cm and b ≈ 8.79 cm.

Example Problem 2: AAS (Angle-Angle-Side)

Problem: A triangle has angles A = 35° and B = 70°, and side a = 8 inches. Find the lengths of sides b and c.

Solution:

1. Identify the knowns: We have A = 35°, B = 70°, and a = 8 inches.

2. Find the third angle: C = 180° - A - B = 180° - 35° - 70° = 75°

3. Apply the Law of Sines: Again, we'll use ratios with the known side and its opposite angle:

   a/sin A = b/sin B

   8/sin 35° = b/sin 70°

   b = (8 * sin 70°) / sin 35°

   b ≈ 12.94 inches

   To find side c, we can use another ratio:

   a/sin A = c/sin C

   8/sin 35° = c/sin 75°

   c = (8 * sin 75°) / sin 35°

   c ≈ 13.25 inches

Therefore, b ≈ 12.94 inches and c ≈ 13.25 inches.

Example Problem 3: Ambiguous Case (SSA)

The ambiguous case (SSA) arises when you know two sides and the angle opposite one of them. This situation can lead to zero, one, or two possible triangles. Let's illustrate:

Problem: A triangle has side a = 12, side b = 15, and angle A = 45°. Find angle B.

Solution:

1. Identify the knowns: a = 12, b = 15, A = 45°

2. Apply the Law of Sines:

   a/sin A = b/sin B

   12/sin 45° = 15/sin B

   sin B = (15 * sin 45°) / 12

   sin B ≈ 0.884

Now, here's where the ambiguity comes in. The inverse sine function (sin⁻¹) will only give you one solution, which is approximately 62°. However, remember that the sine function is positive in both the first and second quadrants. Therefore, there's another possible angle B in the second quadrant:

B₂ = 180° - 62° = 118°

This means there are two possible triangles that satisfy the given conditions! One triangle has B ≈ 62°, and the other has B ≈ 118°. To determine which is valid, you need to consider if the sum of angles in the triangle with 118° remains within the 180° limit.

Let’s check the second case: A + B₂ = 45° + 118° = 163°, which means C₂ = 180° - 163° = 17°. This is possible.

Therefore, there are two possible solutions for this triangle.

Example Problem 4: Solving for an Angle

Problem: In a triangle, side a = 7, side b = 9, and angle A = 30°. Find angle B.

Solution:

1. Identify the knowns: a = 7, b = 9, A = 30°

2. Apply the Law of Sines:

   a/sin A = b/sin B

   7/sin 30° = 9/sin B

   sin B = (9 * sin 30°) / 7

   sin B ≈ 0.643

   B = sin⁻¹(0.643) ≈ 39.9°

In this case, only one solution exists because the value of sin B is less than 1, and there is only one angle within 0° to 180° that satisfies this condition.

Explanation of the Scientific Basis

The Law of Sines is derived from the properties of triangles and trigonometric functions. Its foundation lies in the relationship between the area of a triangle and its sides and angles. Consider a triangle with sides a, b, and c, and angles A, B, and C respectively. The area of the triangle can be expressed in three different ways:

• Area = (1/2)ab sin C
• Area = (1/2)ac sin B
• Area = (1/2)bc sin A

Since all three expressions represent the same area, we can equate them:

(1/2)ab sin C = (1/2)ac sin B = (1/2)bc sin A

Dividing through by (1/2)abc, we arrive at the Law of Sines:

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

Frequently Asked Questions (FAQ)

• When can I not use the Law of Sines? The Law of Sines is not suitable when you only know the three sides of a triangle (SSS). In this case, the Law of Cosines should be used.

• What if I get a sin value greater than 1? This indicates there is no such triangle that can exist with the given measurements. Check your input values for errors.

• What if I get two possible values for an angle? This is the ambiguous case (SSA). You need to determine which solution makes sense within the context of a triangle (sum of angles must be 180°).

• Can I use a calculator for the Law of Sines? Yes, absolutely! Scientific calculators are essential for accurately calculating sine values and solving for unknowns. Make sure your calculator is set to the correct angle mode (degrees or radians).

• How do I know which Law to use (Sines or Cosines)? If you have at least one side and its opposite angle, use the Law of Sines. If you have all three sides or two sides and the included angle, use the Law of Cosines.

Conclusion

The Law of Sines is a powerful tool for solving various triangle problems. By understanding its formula, recognizing the different scenarios (ASA, AAS, SSA), and practicing with diverse examples, you'll build confidence and proficiency in solving complex trigonometric problems. Remember to always double-check your work and consider the ambiguous case when applicable. With consistent practice, mastering the Law of Sines will become second nature. Now go forth and solve those triangles!