University:
The University Of Western Ontario
Course:
MATH 3020A | Introduction to Abstract Algebra
Academic year:

2024
Views:

220

Pages:

3
Author:

Kyuwon J.

Triangle Ambiguity: SSA Case Introduction The Ambiguous Case of triangle solving arises when two sides and a non-included angle (SSA) are given. Unlike other cases (e.g., SAS, ASA), the SSA configuration may result in: • No triangle, • One unique triangle, or • Two distinct triangles. The ambiguity is due to the sine function, which has two possible solutions for angles in the range [0◦ , 180◦ ]. — Conditions for Ambiguity in SSA Triangles Given: • A known angle (∠A), • The side opposite this angle (a), • One other side (b). The steps to determine ambiguity are as follows: Step 1: Check the Validity of sin(B) Using the Law of Sines: a b = sin(A) sin(B) Rearranging to solve for sin(B): sin(B) = b · sin(A) a Case 1: If sin(B) > 1, no triangle can be formed. Case 2: If sin(B) ≤ 1, proceed to Step 2. — 1 Step 2: Solve for ∠B Since sin(B) is valid, calculate ∠B1 : ∠B1 = arcsin(sin(B)) Ambiguity: sin(B) has two possible angles: ∠B1 and ∠B2 = 180◦ − ∠B1 • If ∠B1 + ∠A < 180◦ , the second triangle is valid. • Otherwise, only one triangle exists. — Step 3: Determine the Third Angle For each possible triangle: ∠C = 180◦ − ∠A − ∠B Important Note: The sum of the angles in a triangle must always equal 180◦ . — Step 4: Solve for the Remaining Side Use the Law of Sines to find the remaining side (c): a c = sin(C) sin(A) Rearranging for c: c= a · sin(C) sin(A) — Visualizing the Ambiguity To understand why ambiguity occurs, consider the sine function: sin(θ) = sin(180◦ − θ) For the given angle (∠A) and side lengths (a and b): • One possible triangle forms with ∠B1 . • A second triangle forms with ∠B2 = 180◦ − ∠B1 , provided that ∠A + ∠B2 < 180◦ . The two possible configurations are shown below: Triangle 1: ∠A + ∠B1 + ∠C1 = 180◦ Triangle 2: ∠A + ∠B2 + ∠C2 = 180◦ If ∠A + ∠B2 ≥ 180◦ , only the first triangle exists. — 2 Summary of Ambiguity Cases Given: ∠A, a, and b. The following cases can occur: 1. No Solution: sin(B) > 1. The given side lengths and angle cannot form a triangle. 2. One Solution: • If ∠A + ∠B1 ≥ 180◦ , only one triangle exists. 3. Two Solutions: • If sin(B) ≤ 1 and ∠A + ∠B1 < 180◦ , two triangles exist. — Key Takeaways • The Ambiguous Case applies only when an SSA configuration is given. • Always check sin(B) first to ensure the triangle is valid. • Use the two possible solutions for ∠B to determine the number of triangles. • The ambiguity arises because sin(θ) = sin(180◦ − θ). — 3