In this problem, we're dealing with the ambiguous case in trigonometry for determining the number of possible solutions for a triangle with given angle $A = 33^\circ$, side $a = 5$, and varying side $b$.

The ambiguous case arises when we use the Law of Sines in a triangle with two sides and a non-included angle given (SSA condition), and we seek to determine whether there is one solution, two solutions, or no solution for the triangle.

Explanation of Conditions

1. One solution: This occurs when $b$ is just large enough to form a single triangle. Mathematically, if $b = a$ or if $b > \frac{a}{\sin(A)}$, there will be only one triangle.

   • For this question, the correct option for one solution was chosen as $5 \leq b \leq \frac{5}{\sin(33^\circ)}$.

2. Two solutions: This happens when $b$ is greater than the height but less than $\frac{a}{\sin(A)}$, allowing two possible triangles (one acute and one obtuse) to form.

   • Here, the correct option for two solutions was also selected as $5 < b < \frac{5}{\sin(33^\circ)}$.

3. No solution: If $b$ is less than the minimum height, then no triangle can be formed because side $b$ is too short to reach the opposite side.

   • In the image, the option selected for no solution is incorrect. The correct choice should indicate that $b < \frac{5}{\sin(33^\circ)}$.

Summary of Correct Answers

• One solution: $5 \leq b \leq \frac{5}{\sin(33^\circ)}$
• Two solutions: $5 < b < \frac{5}{\sin(33^\circ)}$
• No solution: $b < \frac{5}{\sin(33^\circ)}$ (Note: The choice marked here was incorrect).

Would you like more detailed calculations on these thresholds or have any questions about this solution?

Follow-up Questions

1. How does the ambiguous case apply to different types of triangles?
2. Can we use the Law of Sines when there’s an obtuse angle involved?
3. How do we calculate the height of a triangle in SSA configurations?
4. Why do some SSA configurations yield no solution?
5. How do different angles impact the possibility of one or two solutions?

Tip

In ambiguous case problems, always calculate the height as a reference; it helps in determining the possible number of solutions for SSA triangles.