Law of Sines Lecture Notes

Introduction to the Law of Sines

• In a triangle, the angles are represented by capital letters (A, B, C) and the sides opposite those angles by lowercase letters (a, b, c).
• The Law of Sines formula is:
  [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
• This equation can be used to find missing angles or sides in a triangle.
• The sum of the angles in a triangle equals 180 degrees.

Example 1

• Given:
  • Angle A = 60 degrees
  • Angle B = 70 degrees
  • Side a = 8

Steps to Solve

1. Find Angle C:
   • [ C = 180 - A - B = 180 - 60 - 70 = 50] degrees.
2. Use Law of Sines to find side b:
   • [ \frac{b}{\sin B} = \frac{a}{\sin A} ]
   • Plugging in values:
     [ \frac{b}{\sin 70} = \frac{8}{\sin 60} ]
   • Cross-multiply and solve for b:
     [ b = \frac{8 \cdot \sin 70}{\sin 60} \approx 8.68 ]
3. Use Law of Sines to find side c:
   • [ \frac{a}{\sin A} = \frac{c}{\sin C} ]
   • Plugging in values:
     [ \frac{8}{\sin 60} = \frac{c}{\sin 50} ]
   • Cross-multiply and solve for c:
     [ c = \frac{8 \cdot \sin 50}{\sin 60} \approx 7.07 ]

Example 2 (SSA Triangle)

• Given:
  • Angle A = 42 degrees
  • Side a = 10
  • Side b = 9

Steps to Solve

1. Find Angle B:
   • [ \frac{a}{\sin A} = \frac{b}{\sin B} ]
   • Plugging in values:
     [ \frac{10}{\sin 42} = \frac{9}{\sin B} ]
   • Cross-multiply and solve for ( \sin B ):
     [ 9 \cdot \sin 42 \approx 10 \cdot \sin B ]
   • Find angle B:
   • [ B \approx 37.03 \text{ degrees} ]
   • Check for second solution:
   • Second possible angle: [ 180 - 37.03 = 143 \text{ degrees} ]
   • Check if valid: [ 42 + 143 > 180 ] (not valid)
   • Therefore, angle B = 37 degrees.
2. Find Angle C:
   • [ C = 180 - A - B = 180 - 42 - 37 = 101 \text{ degrees} ]
3. Find Side c:
   • [ \frac{a}{\sin A} = \frac{c}{\sin C} ]
   • Plugging in values:
     [ \frac{10}{\sin 42} = \frac{c}{\sin 101} ]
   • Solve for c:
     [ c \approx 14.67 ]

Example 3 (No Solution)

• Given:
  • Angle A = 75 degrees
  • Side a = 8
  • Side c = 9

Steps to Solve

1. Check for potential angle c:
   • [ \frac{c}{\sin C} = \frac{a}{\sin A} ]
   • If computed ( \sin C ) exceeds 1, triangle has no solution.
   • In this case, ( \sin C ) is out of range, thus the triangle cannot be formed.

Example 4 (Another SSA Triangle)

• Given:
  • Angle A = 30 degrees
  • Side a = 7
  • Side b = 8

Steps to Solve

1. Find Angle B:
   • [ \frac{a}{\sin A} = \frac{b}{\sin B} ]
   • Solve for angle B:
     [ B \approx 34.85 \text{ degrees} ]
   • Find second solution:
     [ 180 - 34.85 \approx 145.15 \text{ degrees} ]
   • Check for valid triangle:
   • First triangle valid; second triangle calculated as well.
2. Find Angle C and Side c for both triangles:
   • First triangle: ( C \approx 115.15 ), side c ( c \approx 12.7 ).
   • Second triangle: ( C \approx 4.85 ), side c ( c \approx 1.2 ).

Summary

• When using inverse sine, always check for multiple solutions, and verify the validity of angles based on their summation to 180 degrees.
• If summation exceeds 180 degrees, discard the second solution.