TR-31: Solving SSA Triangles (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial covers how to solve SSA triangles, which are triangles defined by two sides and a non-included angle. These triangles can present an ambiguous case, leading to situations where there may be 0, 1, or 2 possible solutions. Understanding how to approach these problems is essential for mastering trigonometry in various mathematical contexts, such as exams and practical applications.

Step 1: Understand the SSA Triangle Configuration

• Identify the given values: You will typically have two sides (let's call them a and b) and a non-included angle (angle A).
• Draw the triangle: Sketch the triangle with the given information. Label the sides and the angle clearly.
• Determine the possible scenarios: SSA triangles can lead to:
  • No solution
  • One solution
  • Two solutions

Step 2: Apply the Law of Sines

• Use the Law of Sines formula: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
• Rearrange to find sin B: [ \sin B = \frac{b \cdot \sin A}{a} ]
• Calculate sin B: Substitute your values for a, b, and angle A.

Step 3: Analyze the Value of sin B

• Check the range: The range for sine is between -1 and 1. If sin B is greater than 1 or less than -1, then there are no solutions.
• Calculate angle B:
  • If sin B is valid, use the inverse sine function: [ B = \sin^{-1}(\sin B) ]

Step 4: Determine the Number of Solutions

• Single solution scenario: If angle B gives a value less than 90 degrees, you will have a second potential angle (180 - B).
• Two solutions scenario:
  • If angle B is acute, calculate:
    • ( B_1 = B )
    • ( B_2 = 180 - B )
• No solutions scenario: If sin B is invalid or if the calculated angles lead to a contradiction.

Step 5: Find Remaining Angles and Side Lengths

• Calculate angle C:
  • Use the triangle angle sum property: [ C = 180 - A - B ]
• Use the Law of Sines again to find the remaining side (if needed): [ c = \frac{a \cdot \sin C}{\sin A} ]

Conclusion

In summary, solving SSA triangles requires careful analysis of the given sides and angles using the Law of Sines. By following the steps outlined, you can determine whether there are 0, 1, or 2 solutions for your triangle. Practicing with additional problems will help solidify your understanding of these concepts and prepare you for more advanced trigonometric applications. For further practice, consider reviewing the extra problems and drills linked in the video description.