Understanding Law of Sines and Cosines

May 9, 2025

Lecture Notes: Law of Sines and Law of Cosines

Introduction

  • Law of Sines and Cosines are used when working with non-right triangles (triangles without a 90-degree angle).
  • Law of Sines: Useful for solving triangles when we have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
  • Law of Cosines: Used for triangles when we have three sides (SSS) or two sides and the included angle (SAS).

Law of Sines

  • Formula Options:
    • ( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} )
    • Can also be rearranged to: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )
  • Key Concept: Each angle and its opposite side form a "pair".
  • Solving Process:
    1. Identify pairs (angle and opposite side).
    2. Use any two of the three ratios to form a proportion.
    3. Solve for the unknown variable.

Example 1: Finding an Angle

  • Given a triangle with angles and sides, solve for angle C using known values.
  • Steps:
    1. Set up the proportion: ( \frac{\sin 70^\circ}{7} = \frac{\sin C}{6} )
    2. Solve for ( \sin C ): Multiply both sides by 6.
    3. Use inverse sine to find ( C ): ( C \approx 53.7^\circ )

Example 2: Finding a Side

  • Given angles and sides, solve for side X.
  • Steps:
    1. Set up the equation: ( \frac{\sin 35^\circ}{X} = \frac{\sin 50^\circ}{10} )
    2. Cross-multiply and solve for X.
    3. Compute X: ( X \approx 7.5 )

Law of Cosines

  • Formula:
    • ( C^2 = A^2 + B^2 - 2AB \cdot \cos C )
    • Analogous forms for A and B.
  • Use Cases:
    • When given all three sides (SSS) or side-angle-side (SAS).

Example 3: Solving for a Side

  • Given two sides and included angle, find the unknown side.
  • Formula: ( C^2 = A^2 + B^2 - 2AB \cdot \cos C )
  • Steps:
    1. Substitute known values into the formula.
    2. Solve for ( C ) by taking square root after calculations.
    3. Result: ( C \approx 6.7 )

Example 4: Solving for an Angle

  • Given all three sides, solve for an angle using the law of cosines.
  • Steps:
    1. Use rearranged law of cosines for angle B: ( B^2 = A^2 + C^2 - 2AC \cdot \cos B )
    2. Simplify and isolate ( \cos B ).
    3. Take inverse cosine to solve for ( B ): ( B \approx 96.7^\circ )

Additional Notes

  • Subsequent Topics: Ambiguous cases in the law of sines can result in one triangle, two triangles, or no triangle, which will be discussed further in a following video.

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