Lecture on the Sine Rule and the Ambiguous Case

Introduction to the Sine Rule

• Sine Rule Basics
  • Used to find angles or sides in triangles.
  • Given an angle θ and sides a and c, you can find angle C.
  • Sine Rule Formulas:
    • ( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} )
    • ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )
  • These formulas can be used interchangeably based on the given values.

The Ambiguous Case

• When it Occurs

  • Occurs when using the sine rule to find an angle.
  • Given two sides (one shorter, one longer) and an angle opposite the shorter side.
  • Can result in two possible solutions for an angle.

• Illustrated Example

  • Lengths: a and c (same in both solutions), given angle θ.
  • Two possible angles C:
    • One acute (< 90 degrees).
    • One obtuse (> 90 degrees).

Solving a Triangle Problem

• Problem Setup:

  • Triangle ABC, angle BAC = 40 degrees.
  • Side c = 7, side a = 5.
  • Find angle BCA.

• Using the Sine Rule

  • Set up using ( \frac{\sin BCA}{7} = \frac{\sin 40}{5} ).
  • Solve for ( \sin(BCA) = \frac{7 \times \sin 40}{5} ).
  • Calculate BCA using inverse sine function.

• Calculation

  • Using a calculator, find the acute angle BCA ≈ 64.1 degrees.

Addressing the Ambiguous Case

• Alternative Solution

  • Consider the obtuse angle possibility.
  • For a triangle to be valid, all angles must sum to 180 degrees.

• Finding the Second Angle

  • Reflect original solution to find BCA' = 180 - 64.1 = 115.9 degrees.

• Verification

  • Check if this new angle makes a valid triangle:
    • Angles: 40 + 115.9 + another angle should sum to 180.
    • Confirm that the second configuration satisfies triangle properties.

Conclusion

• Understanding the Ambiguous Case

  • In certain conditions, two distinct angle solutions can result.
  • Both solutions meet the triangle criteria (sum of angles = 180 degrees).
  • Important to verify that both solutions are possible configurations.

• Takeaway

  • When solving for angles with the sine rule, always consider the ambiguous case.
  • This ensures all possible solutions are identified in a problem.