Overview
This lecture explains the ambiguous case in triangle solving using the Sine Law, focusing on when two different triangles can be formed with given side and angle information.
The Ambiguous Case in Triangles
- The ambiguous case occurs when given SSA (Side-Side-Angle) information, which can sometimes produce two valid triangles.
- The ambiguous case is only possible if the given angle is acute and the given side lengths meet certain criteria.
Conditions for the Ambiguous Case
- If the side opposite the given angle (a) is longer than the side adjacent to the angle (b), only one triangle is possible.
- If the opposite side (a) is shorter than the height from the angle but greater than the height, and also less than the adjacent side (b), two triangles are possible.
- If the opposite side (a) is shorter than the height, no triangle is possible.
Using the Sine Law to Solve the Ambiguous Case
- The Sine Law: sin(A)/a = sin(B)/b.
- Calculate the height (h) using h = b × sin(A).
- If a > h and a < b, proceed to find possible triangles.
- Calculate angle B for the first triangle with sin(B) = (b × sin(A)) / a.
- Find the second possible angle by calculating B' = 180° – B.
- Verify the triangle is valid by ensuring the sum of the angles is less than 180°.
Example Problem Process
- Identify the height using trigonometry: h = b × sin(A).
- Check if a > h and a < b to confirm the ambiguous case.
- Use the Sine Law to solve for possible angles.
- Determine both possible configurations for the triangle.
- Confirm both angles are valid by checking their sum with the given angle is less than 180°.
Key Terms & Definitions
- Ambiguous case — Situation in triangle solving (SSA) where two different triangles can satisfy the given information.
- Sine Law — Equation relating sides and angles in a triangle: sin(A)/a = sin(B)/b.
- Height (h) — Perpendicular distance from the angle to the side opposite, calculated as h = b × sin(A).
Action Items / Next Steps
- Practice solving more ambiguous case problems using the Sine Law.
- Review previous materials on determining triangle existence if not yet completed.