TR-30: SSA Triangles Overview (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial provides an overview of the SSA (Side-Side-Angle) triangle configuration in trigonometry, often referred to as the ambiguous case. In this scenario, two sides and a non-included angle are known, which can lead to 0, 1, or 2 possible triangle solutions. Understanding this case is crucial for solving various trigonometric problems, particularly in exams like those from Edexcel or CIE.

Step 1: Understanding the Ambiguous Case

• Definition: The SSA triangle configuration occurs when you have two sides of a triangle and an angle that is not between those sides.
• Possibilities: This configuration can lead to:
  • No triangle solutions (0)
  • One unique triangle solution (1)
  • Two possible triangle solutions (2)

Step 2: Identifying Conditions for Solutions

• Use the Law of Sines: To determine the number of possible triangles, apply the Law of Sines:

  [ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

  where:

  • ( a ) and ( b ) are the lengths of the known sides.
  • ( A ) is the known angle.
  • ( B ) is the angle opposite side ( b ).

• Steps to Identify Solutions:

  1. Calculate angle ( B ) using:

     [ \sin(B) = \frac{b \cdot \sin(A)}{a} ]

  2. Check the value of ( \sin(B) ):

     • If ( \sin(B) > 1 ), no triangle exists (0 solutions).
     • If ( \sin(B) = 1 ), one triangle exists (1 solution).
     • If ( 0 < \sin(B) < 1 ), two triangles may exist (2 solutions).

Step 3: Calculating Possible Angles

• Finding Angles:

  • Use the inverse sine function to find angle ( B ):

    [ B = \sin^{-1} \left( \frac{b \cdot \sin(A)}{a} \right) ]

  • If two triangles exist, the second angle ( B' ) can be found using:

    [ B' = 180^\circ - B ]

Step 4: Determining Remaining Angles and Sides

• For Each Triangle:

  • Calculate angle ( C ):

    [ C = 180^\circ - A - B ]

  • Calculate the remaining side ( c ) using the Law of Sines:

    [ \frac{c}{\sin(C)} = \frac{a}{\sin(A)} ]

Conclusion

In summary, when dealing with SSA triangles, it's essential to understand the ambiguous case and apply the Law of Sines effectively. By following the outlined steps, you can determine whether there are 0, 1, or 2 triangle solutions. This knowledge is particularly useful in solving trigonometry problems in various exams. For further practice, consider exploring additional examples and problems related to the SSA case.