Solving Triangles using Law of Cosines and Law of Sines

Triangle with Side A = 10, Side B = 20, and Angle C = 60°

Triangle Type

• Side-Angle-Side (SAS) configuration

Solving Strategy

• Law of Sines cannot be used initially due to lack of corresponding angles or sides.
• Use Law of Cosines.

Law of Cosines Formula

• Formula: ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) )
• Alternate forms:
  • ( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) )
  • ( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) )

Calculation Steps

1. Apply Law of Cosines to find side C:
   • ( c^2 = 10^2 + 20^2 - 2 \cdot 10 \cdot 20 \cdot \cos(60°) )
   • ( 100 + 400 - 400 \cdot 0.5 = 300 )
   • ( c = \sqrt{300} = 10\sqrt{3} \approx 17.32 )
2. Use Law of Sines to find angle B:
   • ( \frac{C}{\sin(C)} = \frac{B}{\sin(B)} )
   • ( \frac{17.32}{\sin(60°)} = \frac{20}{\sin(B)} )
   • Solve for B: ( \text{sine of B} = 1 \rightarrow B = 90° )
3. Determine Angle A using sum of angles in triangle:
   • ( A = 180° - 90° - 60° = 30° )

Result

• The triangle is a right triangle.

Triangle with Side A = 7, Side B = 8, and Side C = 9

Triangle Type

• Side-Side-Side (SSS) configuration

Solving Strategy

• Begin with Law of Cosines to find the first angle.

Calculation Steps

1. Apply Law of Cosines to find Angle C:
   • Formula: ( \cos(C) = \frac{c^2 - a^2 - b^2}{-2ab} )
   • Calculation: ( \cos(C) = \frac{81 - 49 - 64}{-2 \cdot 7 \cdot 8} = \frac{-32}{-112} = 0.2857 )
   • ( C = \text{arc cos}(0.2857) \approx 73.4° )
2. Use Law of Sines to find Angle A:
   • ( \frac{C}{\sin(C)} = \frac{A}{\sin(A)} )
   • Solve: ( \sin(A) = \frac{7 \cdot \sin(73.4°)}{9} \approx 0.7453 )
   • ( A = \text{arc sin}(0.7453) \approx 48.2° )
3. Calculate Angle B:
   • ( B = 180° - 73.4° - 48.2° = 58.4° )

Conclusion

• For SSS, initially apply Law of Cosines to find an angle.
• Use Law of Sines for the remaining angles.

Key Takeaway

• Use Law of Cosines when given SAS or SSS triangles to find angles or sides initially, followed by Law of Sines for remaining calculations if possible.