Law of Sines Summary

  • Law of Sines: For any triangle, the ratio of a side length to the sine of its opposite angle is constant for all three sides.
    • Formula: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
  • Applicable Cases for Law of Sines:
    • SAA or ASA: Two angles and one side.
    • SSA: Two sides and the angle opposite one of the sides.
  • Right Triangle vs. Oblique Triangle:
    • Right triangles utilize Pythagorean theorem and SOH-CAH-TOA.
  • Examples:
    • Example 1: Solve for triangle ΔEFG given sides/angles. Calculation steps involve:
    • Find missing angle: ∠F=180°−(∠E+∠G)\angle F = 180° - (\angle E + \angle G)
    • Use ratios to find other sides: g=4⋅430540=4.74g = \frac{4 \cdot 430}{540} = 4.74, f=4⋅430sin⁡850⋅sin⁡∠Ef = \frac{4 \cdot 430}{\sin 850}\cdot \sin \angle E.
    • Example 2: Solve for triangle ΔMAN with given angles and sides. Main steps include finding angles and using sine relations.
  • Applications:
    • Measuring height of objects (e.g., tree) using angle of elevation and Law of Sines:
    • Calculation: x⋅sin⁡(35°)=100⋅sin⁡(60°)x \cdot \sin(35°) = 100 \cdot \sin(60°) gives height of tree as approximately 66.23ft66.23 ft.
    • Finding distances between points (e.g., tents in camping scenario) based on angles and side lengths:
    • Calculation: x⋅sin⁡(30°)=10⋅sin⁡(105°)x \cdot \sin(30°) = 10 \cdot \sin(105°) gives the distance as approximately 5.18m5.18 m.