Sine Law - Ambiguous Case Interactive Explorer

Explore the ambiguous case of the sine law in solving triangle problems interactively. This visual tool demonstrates when given two sides and a non-included angle (SSA), there can be zero, one, or two possible triangles.

The Sine Law

\frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b} = \frac{\sin(\gamma)}{c}

When solving triangle problems given sides \(a\) and \(b\) and angle \(\alpha\) (with \(\alpha\) acute), we use the sine law to find \(\sin(\beta) = \frac{b \sin(\alpha)}{a}\). The number of solutions depends on this relationship.

Geometric Construction

Side b (AC): 3.0

Angle α (degrees): 30°

Side a (BC): 3.0

Height (h = b sin α): 1.50

Quick Case Selection

No Intersection

a < h

One Intersection

a = h

Two Intersections

h < a < b

One Valid Intersection

a ≥ b

Current Case: Two Solutions (Ambiguous Case)

h < a < b

The circle intersects line l at two points (B and B'), resulting in two possible triangles: triangle ABC and triangle AB'C. This is the ambiguous case.

\[ \sin(\beta) = \frac{b \sin(\alpha)}{a} = \frac{3 \times \sin(30^\circ)}{3} = 0.5 \] Since \(0 < \sin(\beta) < 1\) and \(a < b\), there are two possible angles for \(\beta\): \(\beta = 30^\circ\) or \(\beta = 150^\circ\) (but must satisfy triangle sum of angles).

Step-by-Step Geometric Solution

Draw a line (l) and mark point A on it
Draw segment AC of length b at angle α to line (l)
Draw a circle with center C and radius equal to side a
Locate intersection points of the circle and line (l)

Case 1: No Solution

Condition: \(a < h\) where \(h = b \sin(\alpha)\)

The circle does not intersect line l.

a < b \sin(\alpha)

Case 2: One Solution (Right Triangle)

Condition: \(a = h\)

The circle touches line l at exactly one point.

a = b \sin(\alpha)

Case 3: Two Solutions (Ambiguous Case)

Condition: \(h < a < b\)

The circle intersects line l at two points.

b \sin(\alpha) < a < b

Case 4: One Solution

Condition: \(a \geq b\)

The circle intersects line l at one point (the other intersection is at or to the left of A).

a \geq b

Interactive Tutorial

Use the sliders to set parameters b and α to 3 and 30° respectively if they are not already.
Use the Case buttons above to quickly explore each scenario.
Alternatively, adjust side a manually to explore the four different cases:

Set a < 1.5 for Case 1: No Solution
Set a ≈ 1.5 for Case 2: One Solution (Right Triangle)
Set 1.5 < a < 3 for Case 3: Two Solutions (Ambiguous Case)
Set a ≥ 3 for Case 4: One Solution

Experiment with different values of b and α to see how the conditions change.