Visualizing the Ambiguous Case of the Law of Sines (SSA Diagrams & Examples)

📚 Quick Study Guide

• 📐 The Ambiguous Case arises when using the Law of Sines with Side-Side-Angle (SSA) information. This means you know two sides and an angle opposite one of them.
• 🤔 There can be zero, one, or two possible triangles that satisfy the given conditions.
• 📏 To determine the number of possible triangles, compare the length of the side opposite the given angle ($a$) with the height ($h$) of the triangle and the other given side ($b$). $h = b \cdot \sin(A)$.
• 🔑 Case 1: $a < h$: No triangle exists.
• 🔑 Case 2: $a = h$: One triangle exists (a right triangle).
• 🔑 Case 3: $h < a < b$: Two triangles exist. This is the ambiguous case.
• 🔑 Case 4: $a \ge b$: One triangle exists.
• ✍️ The Law of Sines states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Practice Quiz

1. Given triangle ABC with $A = 30^\circ$, $b = 12$, and $a = 6$, how many possible triangles can be formed?

   1. 0
   2. 1
   3. 2
   4. Cannot be determined

2. In triangle XYZ, $x = 8$, $y = 10$, and $X = 45^\circ$. Find the value of $\sin(Y)$.

   1. $\frac{4\sqrt{2}}{5}$
   2. $\frac{5\sqrt{2}}{4}$
   3. $\frac{\sqrt{2}}{2}$
   4. $\frac{1}{2}$

3. For triangle PQR, $p = 5$, $q = 8$, and $P = 30^\circ$. What is the height, $h$, of the triangle with respect to side $r$?

   1. 4
   2. $4\sqrt{3}$
   3. $\frac{5}{2}$
   4. $\frac{8\sqrt{3}}{2}$

4. Triangle ABC has $A = 60^\circ$, $b = 20$, and $a = 10$. How many possible triangles exist?

   1. 0
   2. 1
   3. 2
   4. Infinitely many

5. Given $\triangle DEF$ with $d = 7$, $e = 9$, and $D = 35^\circ$, is this an example of the ambiguous case?

   1. Yes
   2. No
   3. Cannot be determined
   4. Only if $E > 90^\circ$

6. In $\triangle MNO$, $m = 15$, $n = 10$, and $M = 110^\circ$. How many triangles are possible?

   1. 0
   2. 1
   3. 2
   4. 3

7. For triangle ABC, $a = 4$, $b = 6$, and $A = 20^\circ$. What is the approximate measure of angle B in the first possible triangle?

   1. $30.46^\circ$
   2. $149.54^\circ$
   3. $20^\circ$
   4. $60^\circ$