Law of Cosines Summary

Law of Cosines Overview

  • Use for triangles with side-side-angle (SAS) and side-side-side (SSS) configurations.

Key Equations

  • For side length $a$: a2=b2+c2−2bc⋅cos⁡(A)a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
  • For side length $b$: b2=a2+c2−2ac⋅cos⁡(B)b^2 = a^2 + c^2 - 2ac \cdot \cos(B)
  • For side length $c$: c2=a2+b2−2ab⋅cos⁡(C)c^2 = a^2 + b^2 - 2ab \cdot \cos(C)

Finding Angles

  • To find angle $B$: cos⁡(B)=a2+c2−b22ac\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
  • Remember to use inverse cosine to solve for angle.

Example 1: Finding Angle B

  • Given: $a = 19$, $b = 19\sqrt{2}$, $c = 38$
  • Use: b2=a2+c2−2ac⋅cos⁡(B)b^2 = a^2 + c^2 - 2ac \cdot \cos(B)
  • Calculate:
    • $b^2 = 722$, $a^2 = 361$, $c^2 = 1444$
    • Rearranged equation gives cos⁡(B)=0.75\cos(B) = 0.75
    • Final angle: B≈41.41∘B \approx 41.41^{\circ}

Example 2: Largest Angle in Triangle

  • Given: $a = 50$, $b = 41$, $c = 29$
  • Largest angle across from largest side has angle $A$.
  • Use for $A$: a2=b2+c2−2bc⋅cos⁡(A)a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
  • Calculate:
    • Result yields: A≈89.47∘A \approx 89.47^{\circ}

Example 3: Finding Angle B in SAS

  • Given: Side $b$, Side $c$, and angle $A$.
  • Use: a2=b2+c2−2bc⋅cos⁡(A)a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
  • Find $a$, then use b2=a2+c2−2ac⋅cos⁡(B)b^2 = a^2 + c^2 - 2ac \cdot \cos(B) to find angle $B$.
  • Final angle: B≈31.897∘B \approx 31.897^{\circ}

Important Reminders

  • Use Law of Cosines for SSS and SAS triangles.
  • Check configurations to apply the correct formula.
  • Always verify calculations carefully.