Lecture on Sine Rule in Trigonometry

Introduction

• Right Angle Triangles: Use Pythagoras and SOHCAHTOA.
• Non-Right Angle Triangles: Use Sine Rule or Cosine Rule.

Sine Rule

• Formula for Lengths: ( a/\sin A = b/\sin B )
• Formula for Angles: ( \sin A/a = \sin B/b )
• Applications: Used for finding missing lengths or angles in non-right angled triangles.

Explanation of Sine Rule

• Little letters (a, b): Represent lengths.
• Big letters (A, B): Represent angles.
• Opposites: Little 'a' is opposite big 'A', and little 'b' is opposite big 'B'.

Example Problems

Example 1: Finding a Missing Length

• Given: ( x ) opposite to 85°, 12 opposite to 53°.
• Setup: Use ( x/\sin 85 = 12/\sin 53 ).
• Solution:
  • Multiply by ( \sin 85 ) to isolate ( x ).
  • Calculate ( x = 14.97 ) cm (2 decimal places).

Example 2: Another Length Calculation

• Given: ( x ) opposite to 34°, 15 opposite to 97°.
• Setup: ( x/\sin 34 = 15/\sin 97 ).
• Solution:
  • Calculate ( x = 8.45 ) cm (2 decimal places).

Example 3: Calculating Angle in Triangle

• Given: Opposites known, one angle missing.
• Solution:
  • Use sum of angles in triangle (180°) to find the missing angle.
  • Example calculation: ( x = 7.45 ) cm (2 decimal places).

Advanced Example: Finding an Angle Using Sine Rule

• Problem: Find angle ( \angle ABC ).
• Method:
  • Given: ( x ) opposite 5.2, angle 96° opposite 9.1.
  • Use angle formula ( \sin x/5.2 = \sin 96/9.1 ).
  • Solution involves using the inverse sine (shift sine) to find ( x = 34.63° ) (2 decimal places).

Special Case: Obtuse Angle

• Problem: Angle ( \angle CAB ) is obtuse.
• Solution:
  • Calculate using the sine rule, then determine the obtuse angle by ( 180° - \text{given angle} ).
  • Example: ( x = 104.24° ).

Practice Problems

1. Determine Length BC: Find ( x ) with known opposites.
   • Calculation: ( x = 10.79 ) cm (2 decimal places).
2. Find Angle Value: Use known angles and lengths to find ( x ).
   • Calculation: ( x = 51.78° ) (2 decimal places).

Conclusion

• Sine Rule Application: Essential for calculating lengths and angles in non-right angled triangles.
• Inverse Sine: Used for obtaining angles from sine values.
• Cases of Obtuse Angles: Requires additional step of subtracting from 180° to find the correct angle.