Lecture Notes: Law of Sines and Law of Cosines

Overview

• Law of Sines and Law of Cosines are used when dealing with non-right triangles (triangles without a 90-degree angle).
• They are essential for solving triangles where standard trigonometric functions for right triangles do not apply.

Law of Sines

• Formulas:
  • ( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} )
  • Where capital letters denote angles and lowercase letters denote the sides opposite those angles.
• Example Problems:
  1. Finding an Angle:
     • Given: ( \angle A = 70^\circ, a = 7, b = 6 ).
     • To find ( \angle C ), set up the proportion: ( \frac{\sin 70^\circ}{7} = \frac{\sin C}{6} ).
     • Multiply both sides by 6 to isolate ( \sin C ).
     • Solve for ( \angle C ) using the inverse sine function: ( \angle C \approx 53.7^\circ ).
  2. Finding a Side:
     • Given: ( \angle A = 50^\circ, a = 10, \angle B = 35^\circ ).
     • To find side ( b ): ( \frac{\sin 35^\circ}{b} = \frac{\sin 50^\circ}{10} ).
     • Cross-multiply and solve for ( b ): ( b \approx 7.5 ).

Law of Cosines

• Formulas:
  • ( c^2 = a^2 + b^2 - 2ab \cdot \cos C )
  • Use when given: Side-Side-Side (SSS) or Side-Angle-Side (SAS).
• Example Problems:
  1. Finding a Side (SAS):
     • Given: ( a = 8, b = 10, \angle C = 42^\circ ).
     • Use formula: ( c^2 = 8^2 + 10^2 - 2 \cdot 8 \cdot 10 \cdot \cos(42^\circ) ).
     • Solve for ( c ) by taking the square root: ( c \approx 6.7 ).
  2. Finding an Angle (SSS):
     • Given: ( a = 11, b = 15, c = 9 ).
     • Use formula to find ( \angle B ): ( 15^2 = 11^2 + 9^2 - 2 \cdot 11 \cdot 9 \cdot \cos B ).
     • Isolate ( \cos B ) and solve for ( \angle B ) using the inverse cosine function: ( \angle B \approx 96.7^\circ ).

Additional Concepts

• Ambiguous Case (Law of Sines):
  • Depending on given information, it may result in one, two, or no triangles.
  • Further discussion in a separate video on the ambiguous case of the Law of Sines.