Oblique Triangles

• Definition: A triangle without a right angle is called an oblique triangle.
  • Types:
  • Acute Triangle: All angles are acute.
  • Obtuse Triangle: One angle is obtuse, and the other two angles are acute.

Triangle Properties and Solutions

• Given the lengths of one side and any two other parts (angles or sides) of an oblique triangle, it can be solved.
• Cases for solving oblique triangles:
  • ASA or SAA: Two angles and one side.
  • SSA: Two sides and the angle opposite one of them.

Law of Sines

• Definition: For triangle ABC, with sides of lengths a, b, and c opposite angles A, B, and C, respectively:
  $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Key Concept: The ratio of the length of any side to the sine of the angle opposite it is constant.
• Requires at least one complete ratio and additional information about the triangle to apply.

Examples Using the Law of Sines

Example 1: Solving an Oblique Triangle (SAA)

• Given: A = 31°, C = 76°, c = 19
• Find: B
  • $B = 180° - (76° + 31°) = 73°$
• Law of Sines:
  • $\frac{a}{\sin 31°} = \frac{19}{\sin 76°}$
  • Calculate a and b:
  • $a \approx 10.1$
  • $b \approx 18.7$
• Solution: A = 31°, B = 73°, C = 76°, a ≈ 10.1, b ≈ 18.7, c = 19.

Example 2: Solving an Oblique Triangle (ASA)

• Given: A = 132°, B = 28°, c = 6.8
• Find: C
  • $C = 180° - (132° + 28°) = 20°$
• Law of Sines:
  • $\frac{a}{\sin 132°} = \frac{6.8}{\sin 20°}$
  • Calculate a and b:
  • $a \approx 14.8$
  • $b \approx 9.3$
• Solution: A = 132°, B = 28°, C = 20°, a ≈ 14.8, b ≈ 9.3, c = 6.8.

Ambiguous Case of Law of Sines (SSA)

• Scenarios based on side a's length relative to altitude h:
  1. No triangle.
  2. One right triangle.
  3. One oblique triangle.
  4. Two triangles (one acute, one obtuse).

Example 3: One Solution (SSA)

• Given: A = 57°, a = 62, b = 50
• Use Law of Sines to solve:
  • $\frac{\sin A}{a} = \frac{\sin B}{b}$

Example 4: No Solution (SSA)

• Given: C = 39°, c = 6, a = 15

Example 5: Two Solutions (SSA)

• Given: A = 29°, a = 7, b = 10
• Find: Two possible B values, leading to different C calculations.

Area of a Triangle (SAS)

• Formula: Given two sides and the included angle:
  $\text{Area} = \frac{1}{2}ab \sin C$
• Example: A triangle with sides of lengths 5 cm and 8 cm, and an angle of 121°:
  • Area = $\frac{1}{2} \times 5 \times 8 \sin(121°) \approx 17.1 cm²$

Applications of the Law of Sines

• Finding Height: For example, to find the height of a hill given angles of elevation from two points 480 m apart:
  • Angles observed: 42.1° and 32°.
  • Use appropriate triangle calculations to find the height.

Practice Exercises

• Further problem sets to practice solving triangles and applying the Law of Sines in various scenarios.