Solving Triangles Using Law of Cosines and Law of Sines

Introduction

• When given a triangle with different side and angle measures, specific laws in trigonometry can be used to find missing sides or angles.
• Key laws: Law of Cosines and Law of Sines.

Example 1: Solving a Triangle with Given Sides and Angle

• Given:
  • Side a = 10
  • Side b = 20
  • Angle C = 60 degrees
• Type of Triangle: Side-Angle-Side (SAS)

Using Law of Cosines

• Formula: [ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) ]
• Application:
  • Substitute: a = 10, b = 20, C = 60°
  • [ c^2 = 10^2 + 20^2 - 2(10)(20) \cdot \cos(60°) ]
  • [ c^2 = 100 + 400 - 400(0.5) ]
  • [ c^2 = 300 ]
  • [ c = \sqrt{300} = 10\sqrt{3} \approx 17.32 ]

Using Law of Sines

• To Find Angle B
• Formula: [ \frac{c}{\sin(C)} = \frac{b}{\sin(B)} ]
• Application:
  • [ \frac{17.32}{\sin(60)} = \frac{20}{\sin(B)} ]
  • Solve for sin(B): [ \sin(B) = 1 ]
  • Angle B = arcsin(1) = 90°

Finding Angle A

• Angle A = 180° - 90° - 60° = 30°
• Result: The triangle is a right triangle.

Example 2: Solving a Triangle with All Three Sides

• Given:
  • Side a = 7
  • Side b = 8
  • Side c = 9
• Type of Triangle: Side-Side-Side (SSS)

Using Law of Cosines

• Finding Angle C
• Formula: [ \cos(C) = \frac{c^2 - a^2 - b^2}{-2ab} ]
• Application
  • [ \cos(C) = \frac{81 - 49 - 64}{-2(7)(8)} ]
  • [ \cos(C) = \frac{-32}{-112} = \frac{2}{7} \approx 0.2857 ]
  • Angle C = arccos(0.2857) \approx 73.4°

Using Law of Sines

• Finding Angle A
• Formula: [ \frac{c}{\sin(C)} = \frac{a}{\sin(A)} ]
• Application:
  • [ \frac{9}{\sin(73.4)} = \frac{7}{\sin(A)} ]
  • Solve for sin(A): [ \sin(A) \approx 0.7453 ]
  • Angle A = arcsin(0.7453) \approx 48.2°

Finding Angle B

• Angle B = 180° - 73.4° - 48.2° \approx 58.4°

Conclusion

• Using Laws:
  • Use the Law of Cosines first to solve for angles when given all sides (SSS) or two sides and an included angle (SAS).
  • Then apply the Law of Sines for remaining angles or sides.
  • These methods provide a systematic approach to solving any triangle with given measurements.