Solve Each Triangle Round To The Nearest Tenth

Solving Triangles: A Comprehensive Guide to Finding Missing Sides and Angles

Solving a triangle means finding the lengths of all its sides and the measures of all its angles. This seemingly simple task is fundamental to many fields, including surveying, navigation, engineering, and computer graphics. While the methods vary depending on the information given, the underlying principles remain consistent. This comprehensive guide will walk you through different scenarios, providing step-by-step solutions and highlighting key concepts to help you master the art of triangle solving. We'll round all answers to the nearest tenth, as requested.

Understanding the Tools: Laws of Sines and Cosines

Before we dive into specific examples, let's review the two crucial tools used for solving triangles: the Law of Sines and the Law of Cosines.

The Law of Sines

The Law of Sines states the relationship between the angles and the lengths of the sides opposite those angles in any triangle:

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

Where:

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

This law is particularly useful when you know:

• Two angles and one side (AAS or ASA): Given two angles and a side, you can find the third angle (since the angles in a triangle add up to 180°). Then, use the Law of Sines to find the remaining sides.
• Two sides and an angle opposite one of them (SSA): This case, often called the ambiguous case, can lead to zero, one, or two possible triangles. We'll explore this in detail later.

The Law of Cosines

The Law of Cosines provides a relationship between the lengths of all three sides and one angle of a triangle:

a² = b² + c² - 2bc * cos(A)

b² = a² + c² - 2ac * cos(B)

c² = a² + b² - 2ab * cos(C)

This law is crucial when you know:

• Three sides (SSS): Given all three sides, you can use the Law of Cosines to find any of the angles.
• Two sides and the included angle (SAS): Given two sides and the angle between them, you can use the Law of Cosines to find the third side. Then, use the Law of Sines (or the Law of Cosines again) to find the remaining angles.

Solving Triangles: Step-by-Step Examples

Now let's work through various scenarios, applying the Law of Sines and the Law of Cosines to solve for the missing sides and angles. Remember, we'll round all answers to the nearest tenth.

Example 1: AAS (Two Angles and a Non-Included Side)

Let's say we have a triangle with angle A = 40°, angle B = 60°, and side a = 8 cm.

1. Find angle C:

Since the angles in a triangle sum to 180°, C = 180° - 40° - 60° = 80°

2. Use the Law of Sines to find side b:

b/sin(B) = a/sin(A)

b/sin(60°) = 8/sin(40°)

b = 8 * sin(60°) / sin(40°) ≈ 10.7 cm

3. Use the Law of Sines to find side c:

c/sin(C) = a/sin(A)

c/sin(80°) = 8/sin(40°)

c = 8 * sin(80°) / sin(40°) ≈ 12.2 cm

Therefore, the solution is: A = 40°, B = 60°, C = 80°, a = 8 cm, b ≈ 10.7 cm, c ≈ 12.2 cm

Example 2: SAS (Two Sides and the Included Angle)

Consider a triangle with sides a = 5 cm, b = 7 cm, and angle C = 85°.

1. Use the Law of Cosines to find side c:

c² = a² + b² - 2ab * cos(C)

c² = 5² + 7² - 2 * 5 * 7 * cos(85°)

c² ≈ 60.5

c ≈ 7.8 cm

2. Use the Law of Sines to find angle A:

a/sin(A) = c/sin(C)

sin(A) = a * sin(C) / c

sin(A) = 5 * sin(85°) / 7.8

sin(A) ≈ 0.637

A ≈ 39.6° ≈ 40°

3. Find angle B:

B = 180° - A - C = 180° - 40° - 85° = 55°

Therefore, the solution is: A ≈ 40°, B ≈ 55°, C = 85°, a = 5 cm, b = 7 cm, c ≈ 7.8 cm

Example 3: SSS (Three Sides)

Suppose we have a triangle with sides a = 6 cm, b = 8 cm, and c = 10 cm.

1. Use the Law of Cosines to find angle A:

a² = b² + c² - 2bc * cos(A)

cos(A) = (b² + c² - a²) / (2bc)

cos(A) = (8² + 10² - 6²) / (2 * 8 * 10) = 0.8

A = cos⁻¹(0.8) ≈ 36.9° ≈ 37°

2. Use the Law of Cosines to find angle B:

b² = a² + c² - 2ac * cos(B)

cos(B) = (a² + c² - b²) / (2ac)

cos(B) = (6² + 10² - 8²) / (2 * 6 * 10) = 0.6

B = cos⁻¹(0.6) ≈ 53.1° ≈ 53°

3. Find angle C:

C = 180° - A - B = 180° - 37° - 53° = 90°

Therefore, the solution is: A ≈ 37°, B ≈ 53°, C = 90°, a = 6 cm, b = 8 cm, c = 10 cm

Example 4: The Ambiguous Case (SSA)

The SSA case (two sides and a non-included angle) is the most challenging because it can have zero, one, or two solutions. Let's consider a triangle with a = 10, b = 12, and A = 40°.

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

a/sin(A) = b/sin(B)

sin(B) = b * sin(A) / a

sin(B) = 12 * sin(40°) / 10 ≈ 0.771

There are two possible angles for B: B₁ ≈ 50.5° and B₂ ≈ 180° - 50.5° = 129.5°

2. Case 1: B ≈ 50.5°

C = 180° - A - B₁ = 180° - 40° - 50.5° = 89.5°

c/sin(C) = a/sin(A)

c = a * sin(C) / sin(A) = 10 * sin(89.5°) / sin(40°) ≈ 15.5

3. Case 2: B ≈ 129.5°

C = 180° - A - B₂ = 180° - 40° - 129.5° = 10.5°

c/sin(C) = a/sin(A)

c = a * sin(C) / sin(A) = 10 * sin(10.5°) / sin(40°) ≈ 2.8

Therefore, there are two possible solutions in this case.

Conclusion

Solving triangles is a fundamental skill in various fields. Mastering the Law of Sines and the Law of Cosines, along with understanding the nuances of the ambiguous case, equips you with the tools to tackle diverse problems efficiently and accurately. Remember to always check your work and consider the context of the problem to determine the most plausible solution. Practice makes perfect, so work through various examples to build confidence and solidify your understanding. By following the steps outlined and employing these trigonometric laws, you can confidently solve any triangle, rounding your answers to the nearest tenth as needed.