Solve Each Triangle Round Your Answers To The Nearest Tenth

Solving triangles involves finding the measures of all three angles and the lengths of all three sides. When given certain information about a triangle, we can use trigonometric principles such as the Law of Sines, the Law of Cosines, and basic trigonometric ratios to determine the missing angles and sides, rounding each answer to the nearest tenth.

Understanding the Basics of Triangle Solving

Before diving into specific techniques, it's crucial to understand the basic elements and notations of a triangle. A triangle is typically labeled with vertices A, B, and C. The sides opposite these angles are labeled with corresponding lowercase letters: a, b, and c. The angles are represented by their vertex labels: ∠A, ∠B, and ∠C. Solving a triangle means finding the values of all six of these elements.

Key Concepts and Laws

To solve triangles effectively, we rely on several fundamental trigonometric principles:

• Angle Sum Property: The sum of the angles in any triangle is always 180 degrees (∠A + ∠B + ∠C = 180°).
• Law of Sines: This law states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle:

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

• Law of Cosines: This law relates the lengths of the sides of a triangle to the cosine of one of its angles:
  • a² = b² + c² - 2bc * cos(A)
  • b² = a² + c² - 2ac * cos(B)
  • c² = a² + b² - 2ab * cos(C)
• Trigonometric Ratios (SOH CAH TOA): For right triangles, these ratios relate the angles to the sides:
  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent

Scenarios for Solving Triangles

Depending on the given information, different approaches are required. The common scenarios are:

• ASA (Angle-Side-Angle): Two angles and the included side are known.
• AAS (Angle-Angle-Side): Two angles and a non-included side are known.
• SAS (Side-Angle-Side): Two sides and the included angle are known.
• SSS (Side-Side-Side): All three sides are known.
• SSA (Side-Side-Angle): Two sides and a non-included angle are known (this case can be ambiguous).
• Right Triangles: One angle is 90 degrees, and additional information is provided.

Step-by-Step Guide to Solving Different Triangle Cases

Let's explore how to solve each of these cases with practical examples. Each final answer for side lengths and angle measurements will be rounded to the nearest tenth.

1. ASA (Angle-Side-Angle)

In this case, we are given two angles and the side between them.

Example: Given ∠A = 30°, ∠C = 70°, and side b = 10 cm. Solve the triangle.

Steps:

1. Find the third angle: Use the angle sum property to find ∠B.

   • ∠B = 180° - ∠A - ∠C = 180° - 30° - 70° = 80°

2. Use the Law of Sines to find the missing sides:

   • a/sin(A) = b/sin(B)

   • a/sin(30°) = 10/sin(80°)

   • a = (10 * sin(30°)) / sin(80°)

   • a ≈ (10 * 0.5) / 0.9848 ≈ 5.077

   • a ≈ 5.1 cm (rounded to the nearest tenth)

   • c/sin(C) = b/sin(B)

   • c/sin(70°) = 10/sin(80°)

   • c = (10 * sin(70°)) / sin(80°)

   • c ≈ (10 * 0.9397) / 0.9848 ≈ 9.542

   • c ≈ 9.5 cm (rounded to the nearest tenth)

Solution:

• ∠A = 30°
• ∠B = 80°
• ∠C = 70°
• a ≈ 5.1 cm
• b = 10 cm
• c ≈ 9.5 cm

2. AAS (Angle-Angle-Side)

Here, we are given two angles and a side that is not between them.

Example: Given ∠A = 40°, ∠B = 60°, and side a = 8 inches. Solve the triangle.

Steps:

1. Find the third angle: Use the angle sum property to find ∠C.

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

2. Use the Law of Sines to find the missing sides:

   • b/sin(B) = a/sin(A)

   • b/sin(60°) = 8/sin(40°)

   • b = (8 * sin(60°)) / sin(40°)

   • b ≈ (8 * 0.8660) / 0.6428 ≈ 10.777

   • b ≈ 10.8 inches (rounded to the nearest tenth)

   • c/sin(C) = a/sin(A)

   • c/sin(80°) = 8/sin(40°)

   • c = (8 * sin(80°)) / sin(40°)

   • c ≈ (8 * 0.9848) / 0.6428 ≈ 12.254

   • c ≈ 12.3 inches (rounded to the nearest tenth)

Solution:

• ∠A = 40°
• ∠B = 60°
• ∠C = 80°
• a = 8 inches
• b ≈ 10.8 inches
• c ≈ 12.3 inches

3. SAS (Side-Angle-Side)

In this case, two sides and the included angle are given.

Example: Given a = 5 m, b = 8 m, and ∠C = 75°. Solve the triangle.

Steps:

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

   • c² = a² + b² - 2ab * cos(C)
   • c² = 5² + 8² - 2 * 5 * 8 * cos(75°)
   • c² = 25 + 64 - 80 * 0.2588
   • c² ≈ 89 - 20.704 ≈ 68.296
   • c ≈ √68.296 ≈ 8.264
   • c ≈ 8.3 m (rounded to the nearest tenth)

2. Use the Law of Sines to find one of the missing angles:

   • sin(A)/a = sin(C)/c
   • sin(A)/5 = sin(75°)/8.3
   • sin(A) = (5 * sin(75°)) / 8.3
   • sin(A) ≈ (5 * 0.9659) / 8.3 ≈ 0.5819
   • A ≈ arcsin(0.5819) ≈ 35.6° (rounded to the nearest tenth)

3. Find the third angle: Use the angle sum property.

   • ∠B = 180° - ∠A - ∠C = 180° - 35.6° - 75° = 69.4°

Solution:

• ∠A ≈ 35.6°
• ∠B ≈ 69.4°
• ∠C = 75°
• a = 5 m
• b = 8 m
• c ≈ 8.3 m

4. SSS (Side-Side-Side)

Here, all three sides of the triangle are known.

Example: Given a = 7 ft, b = 9 ft, and c = 12 ft. Solve the triangle.

Steps:

1. Use the Law of Cosines to find one of the angles:

   • cos(A) = (b² + c² - a²) / (2bc)
   • cos(A) = (9² + 12² - 7²) / (2 * 9 * 12)
   • cos(A) = (81 + 144 - 49) / 216
   • cos(A) = 176 / 216 ≈ 0.8148
   • A ≈ arccos(0.8148) ≈ 35.5° (rounded to the nearest tenth)

2. Use the Law of Cosines again to find another angle:

   • cos(B) = (a² + c² - b²) / (2ac)
   • cos(B) = (7² + 12² - 9²) / (2 * 7 * 12)
   • cos(B) = (49 + 144 - 81) / 168
   • cos(B) = 112 / 168 ≈ 0.6667
   • B ≈ arccos(0.6667) ≈ 48.2° (rounded to the nearest tenth)

3. Find the third angle: Use the angle sum property.

   • ∠C = 180° - ∠A - ∠B = 180° - 35.5° - 48.2° = 96.3°

Solution:

• ∠A ≈ 35.5°
• ∠B ≈ 48.2°
• ∠C ≈ 96.3°
• a = 7 ft
• b = 9 ft
• c = 12 ft

5. SSA (Side-Side-Angle) - The Ambiguous Case

This case can be tricky because it might result in zero, one, or two possible triangles. It is called the "ambiguous case" because the given information doesn't always lead to a unique triangle.

Example: Given a = 15 cm, b = 20 cm, and ∠A = 30°. Solve the triangle.

Steps:

1. Use the Law of Sines to find sin(B):

   • sin(B)/b = sin(A)/a
   • sin(B)/20 = sin(30°)/15
   • sin(B) = (20 * sin(30°)) / 15
   • sin(B) ≈ (20 * 0.5) / 15 ≈ 0.6667
   • B ≈ arcsin(0.6667) ≈ 41.8° (rounded to the nearest tenth)

2. Check for a second possible angle B: Since sin(x) = sin(180° - x), there might be another possible angle B.

   • B' = 180° - B = 180° - 41.8° = 138.2°

3. Check if the second angle is valid: Determine if A + B' < 180°.

   • 30° + 138.2° = 168.2° < 180°. So, B' is a valid angle.

Thus, we have two possible triangles:

Triangle 1:

• ∠A = 30°

• ∠B ≈ 41.8°

• ∠C = 180° - 30° - 41.8° = 108.2°

• a = 15 cm

• b = 20 cm

  • c/sin(C) = a/sin(A)
  • c/sin(108.2°) = 15/sin(30°)
  • c = (15 * sin(108.2°)) / sin(30°)
  • c ≈ (15 * 0.9502) / 0.5 ≈ 28.506
  • c ≈ 28.5 cm (rounded to the nearest tenth)

Triangle 2:

• ∠A = 30°

• ∠B' ≈ 138.2°

• ∠C' = 180° - 30° - 138.2° = 11.8°

• a = 15 cm

• b = 20 cm

  • c'/sin(C') = a/sin(A)
  • c'/sin(11.8°) = 15/sin(30°)
  • c' = (15 * sin(11.8°)) / sin(30°)
  • c' ≈ (15 * 0.2039) / 0.5 ≈ 6.117
  • c' ≈ 6.1 cm (rounded to the nearest tenth)

Solution: There are two possible solutions:

Triangle 1:

• ∠A = 30°
• ∠B ≈ 41.8°
• ∠C ≈ 108.2°
• a = 15 cm
• b = 20 cm
• c ≈ 28.5 cm

Triangle 2:

• ∠A = 30°
• ∠B' ≈ 138.2°
• ∠C' ≈ 11.8°
• a = 15 cm
• b = 20 cm
• c' ≈ 6.1 cm

6. Right Triangles

In a right triangle, one angle is 90 degrees, simplifying the solving process.

Example: Given ∠C = 90°, a = 6, and b = 8. Solve the triangle.

Steps:

1. Use the Pythagorean Theorem to find the missing side:

   • c² = a² + b²
   • c² = 6² + 8² = 36 + 64 = 100
   • c = √100 = 10

2. Use trigonometric ratios to find the angles:

   • tan(A) = a/b = 6/8 = 0.75
   • A ≈ arctan(0.75) ≈ 36.9° (rounded to the nearest tenth)

3. Find the third angle: Since it's a right triangle, ∠C = 90°.

   • ∠B = 180° - 90° - 36.9° = 53.1°

Solution:

• ∠A ≈ 36.9°
• ∠B ≈ 53.1°
• ∠C = 90°
• a = 6
• b = 8
• c = 10

Common Pitfalls and How to Avoid Them

• Ambiguous Case (SSA): Always check for a second possible triangle when using the Law of Sines in the SSA case.
• Incorrect Mode on Calculator: Ensure your calculator is in the correct mode (degrees or radians).
• Rounding Errors: Avoid rounding intermediate calculations, as it can lead to significant errors in the final answer. Round only the final answer to the nearest tenth.
• Misapplication of Laws: Ensure you are using the correct law based on the given information. Law of Sines for AAS, ASA, and SSA; Law of Cosines for SAS and SSS.
• Forgetting Angle Sum Property: Always double-check that the sum of the angles in your solution is 180°.

Tips for Accuracy

• Draw Diagrams: Sketching the triangle can help visualize the problem and identify the correct approach.
• Label Carefully: Label all sides and angles correctly to avoid confusion.
• Use Exact Values When Possible: Use exact values for trigonometric functions (e.g., sin(30°) = 0.5) when available to minimize rounding errors.
• Double-Check Your Work: Review your calculations and make sure your answers are reasonable given the triangle's properties.
• Consistent Units: Ensure all measurements are in the same units before performing calculations.

Conclusion

Solving triangles is a fundamental skill in trigonometry with diverse applications in fields like surveying, navigation, and engineering. By understanding the Law of Sines, the Law of Cosines, basic trigonometric ratios, and the angle sum property, you can effectively solve any triangle, given sufficient information. Remember to be cautious of the ambiguous case and to always round your final answers to the nearest tenth for accuracy.