Solve The Triangle. Round To The Nearest Tenth

Solve the Triangle: A Comprehensive Guide to Trigonometry and Problem Solving (Rounded to the Nearest Tenth)

Solving a triangle means finding the lengths of all its sides and the measures of all its angles. This seemingly simple task opens the door to a world of trigonometric applications, vital in fields ranging from surveying and engineering to navigation and computer graphics. This article provides a thorough guide to solving triangles, focusing on various scenarios and techniques, with all calculations rounded to the nearest tenth.

Understanding the Fundamentals: Types of Triangles and Key Concepts

Before diving into solving techniques, let's establish a strong foundation. Triangles are classified into three main types based on their angles:

• Acute Triangles: All angles are less than 90 degrees.
• Right Triangles: One angle is exactly 90 degrees.
• Obtuse Triangles: One angle is greater than 90 degrees.

Furthermore, understanding these key concepts is crucial:

• Sides: Denoted by lowercase letters (a, b, c) corresponding to the opposite angles (A, B, C).
• Angles: Denoted by uppercase letters (A, B, C). These angles are measured in degrees.
• Sine (sin), Cosine (cos), and Tangent (tan): These trigonometric functions relate the angles of a right-angled triangle to the ratios of its sides. Remember SOH CAH TOA:
  • Sin A = opposite/hypotenuse
  • Cos A = adjacent/hypotenuse
  • Tan A = opposite/adjacent
• Law of Sines: Relates the sides and angles of any triangle: a/sin A = b/sin B = c/sin C
• Law of Cosines: Useful for solving triangles when you know all three sides or two sides and the included angle:
  • a² = b² + c² - 2bc * cos A
  • b² = a² + c² - 2ac * cos B
  • c² = a² + b² - 2ab * cos C

Solving Right-Angled Triangles

Solving right-angled triangles is often the starting point for understanding triangle solving. We leverage the Pythagorean theorem (a² + b² = c²) and the trigonometric functions (sin, cos, tan).

Example 1: Finding Sides

Let's say we have a right-angled triangle where angle A = 30 degrees, and the hypotenuse (c) = 10. We need to find the lengths of sides a and b.

1. Find side a (opposite to angle A): sin A = a/c sin 30° = a/10 a = 10 * sin 30° = 10 * 0.5 = 5

2. Find side b (adjacent to angle A): cos A = b/c cos 30° = b/10 b = 10 * cos 30° ≈ 10 * 0.866 = 8.7

Therefore, a = 5 and b ≈ 8.7.

Example 2: Finding Angles

Suppose we have a right-angled triangle with sides a = 6 and b = 8. We need to find angles A and B.

1. Find angle A: tan A = a/b = 6/8 = 0.75 A = arctan(0.75) ≈ 36.9°

2. Find angle B: Since the angles in a triangle add up to 180°, B = 180° - 90° - A ≈ 180° - 90° - 36.9° ≈ 53.1°

Therefore, A ≈ 36.9° and B ≈ 53.1°.

Solving Oblique Triangles (Non-Right-Angled Triangles)

Solving oblique triangles requires the Law of Sines and the Law of Cosines. The approach depends on the given information (SSS, SAS, ASA, AAS).

SSS (Side-Side-Side): All three sides are known.

Example 3: Solving with SSS

Let's say a = 7, b = 9, and c = 11. We can use the Law of Cosines to find an angle, and then the Law of Sines to find the remaining angles.

1. Find angle A: a² = b² + c² - 2bc * cos A 7² = 9² + 11² - 2 * 9 * 11 * cos A cos A = (9² + 11² - 7²) / (2 * 9 * 11) ≈ 0.795 A = arccos(0.795) ≈ 37.4°

2. Find angle B (using Law of Sines): b/sin B = a/sin A 9/sin B = 7/sin 37.4° sin B = 9 * sin 37.4° / 7 ≈ 0.786 B = arcsin(0.786) ≈ 51.8°

3. Find angle C: C = 180° - A - B ≈ 180° - 37.4° - 51.8° ≈ 90.8°

Therefore, A ≈ 37.4°, B ≈ 51.8°, and C ≈ 90.8°.

SAS (Side-Angle-Side): Two sides and the included angle are known.

Example 4: Solving with SAS

Let's say a = 5, c = 8, and B = 60°. We use the Law of Cosines to find side b, then the Law of Sines to find the other angles.

1. Find side b: b² = a² + c² - 2ac * cos B b² = 5² + 8² - 2 * 5 * 8 * cos 60° b² = 49 b = 7

2. Find angle A (using Law of Sines): a/sin A = b/sin B 5/sin A = 7/sin 60° sin A = 5 * sin 60° / 7 ≈ 0.616 A = arcsin(0.616) ≈ 38.1°

3. Find angle C: C = 180° - A - B ≈ 180° - 38.1° - 60° ≈ 81.9°

Therefore, b = 7, A ≈ 38.1°, and C ≈ 81.9°.

ASA (Angle-Side-Angle): Two angles and the included side are known. The third angle is found by subtracting the sum of the two known angles from 180°. Then use the Law of Sines to find the remaining sides.

AAS (Angle-Angle-Side): Two angles and a side opposite to one of them are known. Similar to ASA, find the third angle and use the Law of Sines to find the remaining sides.

Ambiguous Case (SSA)

The SSA case (Side-Side-Angle) can be ambiguous; there might be two possible triangles, one triangle, or no triangle at all. Carefully analyze the given information using the Law of Sines to determine the possibilities.

Practical Applications and Conclusion

Solving triangles is far from a purely theoretical exercise. Its applications span various fields:

• Surveying: Determining distances and areas of land.
• Navigation: Calculating distances and bearings.
• Engineering: Designing structures and calculating forces.
• Computer Graphics: Creating realistic images and animations.
• Physics: Analyzing projectile motion and forces.

Mastering the techniques outlined in this guide—understanding right-angled triangle solutions, applying the Law of Sines and Cosines to oblique triangles, and handling the ambiguous case—will equip you to tackle a wide range of trigonometric problems effectively. Remember to always round your final answers to the nearest tenth, as requested. Practice consistently, and you'll become proficient in this fundamental aspect of mathematics. By understanding the underlying principles and consistently applying the correct formulas, solving triangles becomes an achievable and rewarding skill. This comprehensive guide provides a solid foundation for further exploration into the fascinating world of trigonometry and its practical applications.