Chapter 7 - Applications of Trigonometry and Vectors - Section 7.2 The Ambiguous Case of the Law of Sines - 7.2 Exercises - Page 310: 3

(a) If $4 \lt L \lt 5$, we can draw two triangles. (b) If $L = 4$ or $L \gt 5$, we can draw exactly one triangle. (c) If $L \lt 4$, the line is not long enough to meet the x-axis, so we can draw no triangle.

If the line has length $4$, we can draw a vertical line straight down to the x-axis. If $L \gt 4$, we could draw two lines from the point down to meet the x-axis. However, if $L \gt 5$, one of these two lines will meet the negative x-axis. (a) If $4 \lt L \lt 5$, we can draw two triangles. (b) If $L = 4$ or $L \gt 5$, we can draw exactly one triangle. (c) If $L \lt 4$, the line is not long enough to meet the x-axis, so we can draw no triangle.