Law Of Sines Worksheet And Answers

Law of Sines Worksheet and Answers: A Comprehensive Guide

The Law of Sines is a crucial concept in trigonometry, providing a powerful tool for solving oblique triangles—triangles that don't contain a right angle. This comprehensive guide will delve into the Law of Sines, providing a detailed explanation, practice worksheets with solutions, and tips for mastering this essential trigonometric principle.

Understanding the Law of Sines

The Law of Sines states the relationship between the sides and angles of any triangle. It asserts that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. Mathematically, it's expressed as:

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

Where:

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

This formula is remarkably versatile, allowing us to solve for missing sides or angles when given sufficient information. However, it's important to understand the different scenarios you might encounter and the appropriate approach for each.

When to Use the Law of Sines

The Law of Sines is particularly useful in the following situations:

• ASA (Angle-Side-Angle): When you know two angles and the included side.
• AAS (Angle-Angle-Side): When you know two angles and a non-included side.
• SSA (Side-Side-Angle): When you know two sides and the angle opposite one of them. This case is particularly important because it can lead to ambiguous solutions (meaning there might be two possible triangles that satisfy the given conditions).

Solving Triangles Using the Law of Sines: A Step-by-Step Approach

Let's illustrate the application of the Law of Sines with a step-by-step example. Consider a triangle with:

• Angle A = 40°
• Angle B = 60°
• Side a = 8

Step 1: Find the third angle.

Since the sum of angles in a triangle is always 180°, we can find angle C:

C = 180° - A - B = 180° - 40° - 60° = 80°

Step 2: Apply the Law of Sines to find side b.

We can use the Law of Sines to find side b:

a/sin A = b/sin B

8/sin 40° = b/sin 60°

Solving for b:

b = (8 * sin 60°) / sin 40° ≈ 10.96

Step 3: Apply the Law of Sines to find side c.

Similarly, we can find side c:

a/sin A = c/sin C

8/sin 40° = c/sin 80°

Solving for c:

c = (8 * sin 80°) / sin 40° ≈ 12.26

Therefore, the solution for the triangle is:

• A = 40°
• B = 60°
• C = 80°
• a = 8
• b ≈ 10.96
• c ≈ 12.26

The Ambiguous Case (SSA): Navigating Multiple Solutions

The SSA case (Side-Side-Angle) is unique because it can yield either one, two, or no solutions. This ambiguity arises because the given information might allow for the construction of two different triangles.

To determine the number of solutions, consider the height (h) of the triangle from the vertex opposite the given angle:

h = b * sin A

• If a < h: No solution exists. The given side is too short to form a triangle.
• If a = h: One solution exists (a right-angled triangle).
• If h < a < b: Two solutions exist.
• If a ≥ b: One solution exists.

Let's illustrate the ambiguous case with an example. Consider a triangle with:

• a = 10
• b = 12
• A = 45°

Step 1: Calculate the height (h).

h = b * sin A = 12 * sin 45° ≈ 8.49

Since h < a < b, we expect two solutions.

Step 2: Solve for angle B using the Law of Sines.

a/sin A = b/sin B

10/sin 45° = 12/sin B

sin B = (12 * sin 45°) / 10 ≈ 0.8485

This gives two possible values for B:

B₁ ≈ 58° B₂ ≈ 180° - 58° ≈ 122°

Step 3: Solve for the remaining angles and sides for each solution.

For solution 1 (B₁ ≈ 58°):

C₁ = 180° - A - B₁ ≈ 180° - 45° - 58° ≈ 77°

c₁ = (a * sin C₁) / sin A ≈ (10 * sin 77°) / sin 45° ≈ 13.56

For solution 2 (B₂ ≈ 122°):

C₂ = 180° - A - B₂ ≈ 180° - 45° - 122° ≈ 13°

c₂ = (a * sin C₂) / sin A ≈ (10 * sin 13°) / sin 45° ≈ 3.16

Therefore, there are two possible triangles satisfying the given conditions.

Law of Sines Worksheet: Practice Problems

Here are some practice problems to solidify your understanding of the Law of Sines. Remember to show your work and pay close attention to the ambiguous case (SSA).

Problem 1:

Solve triangle ABC, given:

• A = 30°
• B = 70°
• a = 5

Problem 2:

Solve triangle ABC, given:

• A = 40°
• b = 10
• a = 8

Problem 3:

Solve triangle ABC, given:

• a = 7
• b = 9
• A = 48°

Problem 4:

Solve triangle ABC, given:

• B = 110°
• C = 40°
• b = 12

Problem 5:

Solve triangle ABC, given:

• a = 15
• b = 12
• A = 60°

Law of Sines Worksheet: Answers

Problem 1:

C = 80°, b ≈ 9.64, c ≈ 9.85

Problem 2:

Two solutions:

Solution 1: B₁ ≈ 58°, C₁ ≈ 82°, c₁ ≈ 12.21 Solution 2: B₂ ≈ 122°, C₂ ≈ 18°, c₂ ≈ 3.94

Problem 3:

Two solutions:

Solution 1: B₁ ≈ 68°, C₁ ≈ 64°, c₁ ≈ 6.96 Solution 2: B₂ ≈ 112°, C₂ ≈ 20°, c₂ ≈ 2.84

Problem 4:

A = 30°, a ≈ 6.21, c ≈ 8.03

Problem 5:

One solution: B ≈ 41°, C ≈ 79°, c ≈ 16.7

Remember to always check for the ambiguous case and potential multiple solutions. Practice regularly and gradually increase the complexity of the problems to master this vital trigonometric concept. Understanding the Law of Sines is fundamental for solving a wide variety of problems in geometry, engineering, and other fields. By applying the principles outlined above and practicing with the provided worksheets, you’ll develop a strong foundation in this essential area of trigonometry.