The "ambiguous case" of the Sine Law occurs when solving for a triangle using two sides and a non-included angle (SSA configuration). This situation can lead to three different outcomes:

1. No Triangle: Sometimes, no valid triangle can be formed.
2. One Triangle: There is exactly one possible triangle.
3. Two Triangles: There are two distinct possible triangles.

Understanding the Ambiguous Case:

Suppose you are given:

• Side $a$
• Side $b$
• Angle $A$ (opposite to side $a$)

You want to determine angle $B$ using the Sine Law: $\frac{a}{\sin A} = \frac{b}{\sin B}$ This leads to: $\sin B = \frac{b \cdot \sin A}{a}$

Outcomes Based on the Value of $\sin B$:

1. No Triangle (Impossible Case):

   • If $\frac{b \cdot \sin A}{a} > 1$, then $\sin B > 1$, which is impossible since the sine of an angle cannot exceed 1. Therefore, no triangle can be formed.

2. One Triangle:

   • If $\frac{b \cdot \sin A}{a} = 1$, then $\sin B = 1$, which means $B = 90^\circ$. In this case, only one right triangle can be formed.

   • Alternatively, if $0 < \frac{b \cdot \sin A}{a} < 1$ and side $a$ is longer than side $b$, only one triangle can be formed with $B = \sin^{-1} \left(\frac{b \cdot \sin A}{a}\right)$.

3. Two Triangles:

   • If $0 < \frac{b \cdot \sin A}{a} < 1$ and side $a$ is shorter than side $b$, two different angles $B$ can satisfy the equation:
     • $B_1 = \sin^{-1} \left(\frac{b \cdot \sin A}{a}\right)$
     • $B_2 = 180^\circ - B_1$

   In this case, two distinct triangles can be formed, because both $B_1$ and $B_2$ are valid angles that add up with $A$ to form a triangle.

Example Problem:

Given:

• $A = 30^\circ$
• $a = 7$ units
• $b = 10$ units

Find angle $B$ using the Sine Law: $\sin B = \frac{b \cdot \sin A}{a} = \frac{10 \cdot \sin 30^\circ}{7} = \frac{10 \cdot 0.5}{7} = \frac{5}{7} \approx 0.7143$ Now, $B_1 = \sin^{-1}(0.7143) \approx 45.6^\circ$.

Since $0 < \sin B < 1$, there is also a second solution: $B_2 = 180^\circ - B_1 \approx 180^\circ - 45.6^\circ = 134.4^\circ$

So, two triangles are possible:

• One with $B_1 \approx 45.6^\circ$
• One with $B_2 \approx 134.4^\circ$

Do you have any questions or need further clarification?

Here are 5 related questions to deepen your understanding:

1. How does the ambiguous case affect the overall solution to a triangle problem?
2. What are some real-life scenarios where the ambiguous case might occur?
3. How do you determine the length of the third side in both possible triangles?
4. Can the ambiguous case occur in right triangles? Why or why not?
5. How would you handle the ambiguous case if angle $B$ were obtuse in the first solution?

Tip: When dealing with the ambiguous case, always sketch the potential triangles to visualize the situation better. This can help in understanding whether one or two triangles are possible.