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Understanding Triangle SSA Ambiguity

May 13, 2025

Lecture Notes: Triangle SSA Ambiguity

Introduction

  • Scenario: Given angle A = 58 degrees, side b = 12.8, and side a = 11.4.
  • The triangle is used to illustrate the Side-Side-Angle (SSA) case.

Basic Triangle Setup

  • Label the triangle with angle A at 58 degrees.
  • Recognize that SSA can potentially lead to:
    • Two triangles
    • One triangle
    • No triangle
  • The current case requires exploration of these possibilities.

Understanding SSA Ambiguity

  • Issue: With SSA, the information provided can correspond to more than one triangle configuration.
  • Visualization: Imagine a doorway hinge effect where side a can swing to form different triangles.
  • Possibilities:
    • Two possible triangles
    • Triangles that might not form at all if side lengths do not meet.

Solving for Angle B using Law of Sines

  • Law of Sines Formula: ( \frac{a}{\sin A} = \frac{b}{\sin B} )
  • Given: ( \frac{11.4}{\sin 58} = \frac{12.8}{\sin B} )
  • Solve for ( \sin B ):
    • ( \sin B = \frac{12.8 \times \sin 58}{11.4} )
    • ( \sin B = 0.9522 )
    • Use inverse sine to find angle B.
    • Angle B = 72.21 degrees.

Exploring Angle B Solutions

  • Unit Circle Insight: Sine values repeat in two quadrants (1st and 2nd).
  • Two Possible B Values:
    • Primary solution from inverse sine = 72.21 degrees.
    • Secondary solution: ( B = 180 - 72.21 = 107.79 ) degrees.

Evaluating Cases

  • Case 1: ( B = 72.21 ) degrees
    • Remaining angle C: ( C = 180 - 58 - 72.21 = 49.79 ) degrees.
  • Case 2: ( B = 107.79 ) degrees
    • Remaining angle C: ( C = 180 - 58 - 107.79 = 14.21 ) degrees.

Solving for Side C

  • Use the Law of Sines for both cases to find side C:
    • Case 1:
      • ( C = \frac{11.4 \times \sin 49.79}{\sin 58} )
      • ( C = 10.26 )
    • Case 2:
      • ( C = \frac{11.4 \times \sin 14.21}{\sin 58} )
      • ( C = 3.29 )

Key Takeaways

  • Recognize ambiguity in SSA cases leading to multiple triangle possibilities or no triangle.
  • Understand the need to check for obtuse and acute triangles when applying inverse sine.
  • Next steps: Explore scenarios where no triangle exists.

Conclusion

  • Importance of checking potential configurations in SSA problems.
  • Upcoming: Examine cases of non-existent triangles.

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