Solution of Triangles: Case of Ambiguity

The solution of triangles involves finding the unknown sides and angles of a triangle when certain sides and angles are given. One particular scenario in the solution of triangles is known as the "case of ambiguity" or the "ambiguous case." This case arises in the context of oblique triangles, specifically when using the Law of Sines, and it occurs when two different triangles can be formed from the given information.

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be expressed as:

$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$

where (a), (b), and (c) are the lengths of the sides, and (A), (B), and (C) are the opposite angles.

Case of Ambiguity

The case of ambiguity occurs when we are given two sides and a non-included angle (SSA condition) to solve a triangle. In this case, there can be zero, one, or two possible solutions for the triangle.

Conditions for Ambiguity

Given Information	Possible Solutions	Reason for Ambiguity
Side (a), side (b), and angle (A) (SSA)	Zero, one, or two	Depending on the relative sizes of (a), (b), and (\sin A), there may be no solution, one solution, or two solutions.

Determining the Number of Solutions

1. No Solution: If (a < b \sin A), then no triangle exists because side (a) is too short to reach side (b) after swinging it out from angle (A).

2. One Solution: If (a = b \sin A), then there is exactly one solution because side (a) just touches side (b) when swung out from angle (A). This is also the case when (a > b) and (A) is obtuse, as there can only be one such triangle.

3. Two Solutions: If (a > b \sin A) and (a < b), there are two possible solutions. This is because side (a) can form two different triangles by swinging it from angle (A).

Formulas for Ambiguous Case

When there are two possible solutions, we can find the possible values of the unknown angle (B) using the Law of Sines:

$$ \sin B = \frac{b \sin A}{a} $$

If (\sin B) is less than or equal to 1, then there are two possible angles for (B): (B) and (180^\circ - B). We can then use the Law of Sines again to find the corresponding side (c) and angle (C) for each case.

Examples

Let's consider an example to illustrate the case of ambiguity.

Example 1: Two Possible Solutions

Given: (a = 7), (b = 10), and (A = 30^\circ).

1. Check for ambiguity: Since (a < b) and (a > b \sin A), there are two possible solutions.

2. Find (\sin B):

$$ \sin B = \frac{b \sin A}{a} = \frac{10 \sin 30^\circ}{7} \approx 0.714 $$

1. Since (\sin B < 1), there are two possible angles for (B):

$$ B \approx \arcsin(0.714) \approx 45.58^\circ $$

or

$$ B \approx 180^\circ - 45.58^\circ \approx 134.42^\circ $$

1. For each value of (B), we can find (C) and (c) using the Law of Sines and the fact that the sum of angles in a triangle is (180^\circ).

Example 2: One Possible Solution

Given: (a = 12), (b = 10), and (A = 30^\circ).

1. Check for ambiguity: Since (a > b), there is only one possible solution, even though (A) is acute.

2. Find (\sin B):

$$ \sin B = \frac{b \sin A}{a} = \frac{10 \sin 30^\circ}{12} \approx 0.417 $$

1. Since (\sin B < 1), there is only one possible angle for (B):

$$ B \approx \arcsin(0.417) \approx 24.59^\circ $$

1. Find (C) and (c) using the Law of Sines and the angle sum property.

In conclusion, the case of ambiguity in the solution of triangles requires careful analysis of the given information to determine the number of possible solutions. Understanding the conditions and applying the Law of Sines appropriately allows us to resolve the ambiguity and find all potential triangles that satisfy the given conditions.