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Understanding the Law of Sines

Apr 28, 2025

Lecture Notes: The Law of Sines

Introduction

  • Law of Sines Formula:
    • Given a triangle with angles A, B, C and opposite sides a, b, c respectively: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
    • Purpose: To find missing angles or sides in a triangle.

Basic Properties of Triangles

  • Sum of angles in a triangle = 180°.

Example 1: Solving a Triangle

  • Given:
    • Angle A = 60°, Angle B = 70°, Side a = 8
  • Find: Angle C, Side b, Side c

Solution Steps

  1. Find Angle C:
    • A + B + C = 180°
    • C = 180° - 60° - 70° = 50°
  2. Find Side b using Law of Sines:
    • ( \frac{b}{\sin B} = \frac{a}{\sin A} )
    • Solve for b: ( b = \frac{8 \times \sin 70°}{\sin 60°} \approx 8.68 )
  3. Find Side c:
    • ( \frac{c}{\sin C} = \frac{a}{\sin A} )
    • Solve for c: ( c = \frac{8 \times \sin 50°}{\sin 60°} \approx 7.07 )

Example 2: Side-Side-Angle (SSA) Triangle Case

  • Given:
    • Angle A = 42°, Side a = 10, Side b = 9
  • Find: Angle B, Angle C, Side c

Solution Steps

  1. Find Angle B:
    • Solve ( \sin B = \frac{9 \times \sin 42°}{10} \approx 0.602 )
    • B = ( \arcsin(0.602) \approx 37.03° )
    • Possible second solution: 180° - 37.03° = 143° (Check if valid)
    • A + B cannot exceed 180°, 42° + 143° > 180°, so only one solution.
  2. Find Angle C:
    • C = 180° - A - B = 180° - 42° - 37° = 101°
  3. Find Side c:
    • ( c = \frac{10 \times \sin 101°}{\sin 42°} \approx 14.67 )

Example 3: No Solution Case

  • Given:
    • Angle A = 75°, Side a = 8, Side c = 9
  • Issue: Solve ( \sin C = \frac{9 \times \sin 75°}{8} > 1 )
    • No solution as ( \sin C ) cannot exceed 1.

Example 4: Two Solutions

  • Given:
    • Angle A = 30°, Side a = 7, Side b = 8
  • Find: Angle B, Angle C, Side c with two possible triangles.

Solution Steps

  1. Find Angle B:
    • ( \sin B = \frac{8 \times \sin 30°}{7} \approx 0.571 )
    • B = ( \arcsin(0.571) \approx 34.85° )
    • Second potential angle: 180° - 34.85° = 145.15°
    • Check validity: A + B < 180° for both cases
  2. Complete Triangles:
    • First triangle: C = 180° - 30° - 34.85° = 115.15°
    • Second triangle: C = 180° - 30° - 145.15° = 4.85°
  3. Find Side c for each triangle:
    • First triangle: ( c \approx 12.7 )
    • Second triangle: ( c \approx 1.2 )

Key Takeaways

  • Always verify triangle properties (sum of angles, side-angle relationships).
  • SSA triangles may have one, two, or no solutions.
  • Cross-check with triangle inequalities to ensure plausible solutions.

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