Notes on the Law of Cosines and Applications

Law of Cosines

• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• Useful for solving oblique triangles with two sides and the included angle (SAS) or three sides (SSS).

Formulas

• For triangle ABC with sides a, b, c and angles A, B, C:
  • $a^2 = b^2 + c^2 - 2bc \cdot \cos A$
  • $b^2 = a^2 + c^2 - 2ac \cdot \cos B$
  • $c^2 = a^2 + b^2 - 2ab \cdot \cos C$

Application Examples

1. Example with Two Planes:

   • Given two planes crossing at an angle: 49°, distances from crossing: 32 miles and 76 miles.
   • The distance between the planes calculated using the Law of Cosines results in approximately 60.07 miles.

2. Distance between Schools:

   • Distance from Falcon School to Hope Integrated is approximately 220m, from Mother of Perpetual Help School to Falcon School is 340m with a 40° angle.
   • Calculate using the Law of Cosines can give the distance when traveling from one to the other.

3. Tree House Example:

   • Two supporting beams measure 5m and 7m with a 60° angle between them:
   • Diagonal length calculated shows approx 6.24m.

Steps for Solving SAS Case

1. Find the side opposite the given angle using the Law of Cosines.
2. Find the second angle using the Law of Cosines.
3. Calculate the third angle by subtracting from 180°.

Important Considerations

• Three pieces of information are needed for the Law of Cosines:
  • Three sides (SSS)
  • Two sides and an included angle (SAS)

Assignment Reminder

• Solve for missing parts of given triangles, showing solutions on half a sheet of paper.