Lecture on the Law of Sines

Introduction to the Law of Sines

• The Law of Sines is used to find missing angles or sides in a triangle.
• Formula: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
  • a, b, c: side lengths opposite to angles A, B, C respectively.
  • A, B, C: angles of the triangle.
• Note: The sum of angles in a triangle is always 180 degrees.

Example 1: Solving a Triangle with Given Angles and Side

• Given:
  • Angle A = 60 degrees
  • Angle B = 70 degrees
  • Side a = 8
• Find:
  • Angle C, Side b, Side c

Steps to Solve:

1. Find Angle C:

   • Sum of angles: ( A + B + C = 180 )
   • ( C = 180 - 60 - 70 = 50 ) degrees

2. Find Side b Using Law of Sines:

   • Formula: ( \frac{b}{\sin B} = \frac{a}{\sin A} )
   • Substitute and solve: ( b = \frac{8 \cdot \sin 70}{\sin 60} \approx 8.68 )

3. Find Side c:

   • Formula: ( \frac{c}{\sin C} = \frac{a}{\sin A} )
   • Substitute and solve: ( c = \frac{8 \cdot \sin 50}{\sin 60} \approx 7.07 )

Example 2: Side-Side-Angle (SSA) Triangle

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

Steps:

1. Find Angle B:

   • Use Law of Sines: ( \frac{b}{\sin B} = \frac{a}{\sin A} )
   • Solve for B: ( \sin B = \frac{9 \sin 42}{10} = 0.602 )
   • ( B = \text{arc sine}(0.602) \approx 37 ) degrees

2. Check for Second Solution:

   • Calculate supplementary angle: ( 180 - 37 = 143 )
   • Validate angle: ( 42 + 143 > 180 ), only one triangle possible.

3. Find Angle C:

   • ( C = 180 - 42 - 37 = 101 ) degrees

4. Find Side c:

   • Formula: ( \frac{c}{\sin C} = \frac{a}{\sin A} )
   • Solve for c: ( c = \frac{10 \sin 101}{\sin 42} \approx 14.67 )

Example 3: SSA Triangle with No Solution

• Given:
  • Angle A = 75 degrees
  • Side a = 8
  • Side c = 9
• Calculation:
  • ( c \sin A = a \sin C ) leads to invalid sine range. No solution.

Example 4: Two Possible Triangles

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

Steps:

1. Find Angle B:

   • Solve for B: ( B = \text{arc sine}(\frac{4}{7}) \approx 34.9 ) degrees

2. Check for Second Solution:

   • Calculate supplementary angle: ( 180 - 34.9 = 145.1 )
   • Validate angles: ( 30 + 145.1 < 180 ), two triangles possible.

3. Calculate Angle C for both triangles:

   • For first triangle: ( C = 115.1 ) degrees
   • For second triangle: ( C = 4.9 ) degrees

4. Find Side c for Both Triangles:

   • First Triangle: ( c \approx 12.7 )
   • Second Triangle: ( c \approx 1.2 )

Key Takeaways:

• Always check if two solutions are possible in SSA scenarios.
• When angles are supplementary, ensure the sum of angles is less than 180 to validate two triangles.
• Cross-multiplying and using inverse trigonometric functions are crucial in solving for unknowns.