Law of Cosines

Law of Cosines

• The Law of Cosines is used to find a side of a triangle when two sides and the included angle are known.
• It can also find angles when all three sides are known.

Formula:

• c² = a² + b² − 2ab cos(C)
  • Where:
  • c = length of the side opposite angle C
  • a and b = lengths of the other two sides
  • C = angle opposite side c

Key Concepts:

1. Application:

   • Use this formula when you have either:
     • Two sides and the included angle (A), to find the length of the third side (c).
     • All three sides (a, b, c) to find an angle (C).

2. Rounding Answers:

   • Always round answers to the nearest tenth for precision in measurement.

Example Problem:

• If you know:

  • a = 5
  • b = 7
  • C = 60 degrees

• You want to find c:

  1. Substitute values into the formula:
     • c² = 5² + 7² - 2(5)(7)cos(60)
  2. Calculate:
     • c² = 25 + 49 - 70(0.5)
     • c² = 74 - 35
     • c² = 39
     • c = √39 ≈ 6.2

Example of Finding an Angle:

• If a = 5, b = 6, c = 7, find angle C.
  1. Rearrange the formula to solve for cos(C):
     • cos(C) = (a² + b² - c²) / (2ab)
  2. Substitute values:
     • cos(C) = (5² + 6² - 7²) / (2(5)(6))
     • cos(C) = (25 + 36 - 49) / 60
     • cos(C) = 12 / 60 = 0.2
  3. Find angle C:
     • C = cos⁻¹(0.2) ≈ 78.5 degrees

Important Notes:

• Ensure to understand when to apply the Law of Cosines versus the Law of Sines, as they are used in different situations of triangle calculations.