Comprehensive Guide to Solving Triangles and the Ambiguous SSA Case

Solving Triangles and the Ambiguous SSA Case

• The Side-Side-Angle ($SSA$) problem occurs when given the lengths of two sides and the measure of a non-included angle.
• Anytime an $SSA$ condition is presented, an automatic check for multiple potential triangles must be performed.
• Determining the number of possible triangles relies on the value of \n\sin(B)\n obtained via the Law of Sines:
  • If \n\sin(B) > 1\n: The calculation is impossible; no triangle exists ($0$ triangles).
  • If \n\sin(B) = 1\n: Exactly one triangle exists, which is a right-angled triangle ($1$ triangle).
  • If \n\sin(B) < 1\n: At least one triangle exists, with the possibility of a second obtuse triangle ($1$ or $2$ triangles).

Geometric Mechanics of the Swinging Arm

• In an $SSA$ triangle configuration, specific measurements remain fixed:
  • Fixed side length $b$
  • Fixed side length $a$
  • Fixed angle $A$
• Base side $c$ has a fixed starting ray, but its total length is variable prior to solving.
• Side $a$ acts as a swinging arm pivoting from fixed vertex $C$:
  • If side $a$ is shorter than the perpendicular height from vertex $C$ to base side $c$, it cannot reach the base line ($0$ triangles).
  • If side $a$ exactly equals the perpendicular height, it touches base side $c$ at a single point, forming a right angle ($1$ triangle).
  • If side $a$ is longer than the perpendicular height but shorter than side $b$, it swings across base side $c$ and intersects it at two distinct points, creating two distinct valid triangles:
  • One triangle featuring an acute angle for $B_1$
  • One triangle featuring an obtuse angle for $B_2$

Step-by-Step Procedure for Solving the Ambiguous Case

• Step 1: Calculate the Primary Angle $B_1$

  • Use the Law of Sines:     \n\frac{\sin(A)}{a} = \frac{\sin(B)}{b}\n
  • Rearrange to solve for \n\sin(B_1)\n:     \n\sin(B_1) = \frac{b \times \sin(A)}{a}\n
  • Calculate $B_1 = \arcsin(\sin(B_1))$. Standard calculators will strictly return an acute angle ($< 90^\circ$).

• Step 2: Calculate the Primary Third Angle $C_1$

  • Use the internal angle sum rule for triangles ($180^\circ$):     \nC_1 = 180^\circ - A - B_1\n

• Step 3: Calculate Primary Side Length $c_1$

  • Apply the Law of Sines to find side $c_1$:     \n\frac{c_1}{\sin(C_1)} = \frac{b}{\sin(B_1)}\n
  • Rearrange to solve for $c_1$:     \nc_1 = \frac{b \times \sin(C_1)}{\sin(B_1)}\n

• Step 4: Determine the Secondary Obtuse Angle $B_2$

  • Because a straight line equals $180^\circ$, the secondary obtuse angle $B_2$ is the supplementary angle of $B_1$:     \nB_2 = 180^\circ - B_1\n

• Step 5: Calculate the Secondary Third Angle $C_2$

  • Subtract the fixed angle $A$ and the new obtuse angle $B_2$ from $180^\circ$:     \nC_2 = 180^\circ - A - B_2\n

• Step 6: Calculate Secondary Side Length $c_2$

  • Apply the Law of Sines using the secondary angle values $C_2$ and $B_2$:     \nc_2 = \frac{b \times \sin(C_2)}{\sin(B_2)}\n
  • Consistency Check: Because angle $B_2$ is obtuse ($> 90^\circ$), angle $C_2$ will be smaller than $C_1$, causing side length $c_2$ to be significantly shorter than $c_1$. If $c_2$ evaluates to a value longer than $c_1$, or if $B_2 \le 90^\circ$, an error has occurred.

Worked Examples

• Example 1: Initial Sine Value Evaluation

  • Given an $SSA$ setup, applying the Law of Sines yields:     \n\sin(B) = 0.907\n
  • Because $0.907 < 1$, there is a potential for two triangles.
  • Primary acute angle $B_1$:     \nB_1 = 65^\circ\n
  • Secondary obtuse angle $B_2$:     \nB_2 = 180^\circ - 65^\circ = 115^\circ\n

• Example 2: Full Double-Triangle Calculation

  • Given parameters:
  • Primary acute angle $B_1 = 61^\circ$
  • Calculated angle $C_1 = 94^\circ$
  • Fixed side length $b = 30.6$
  • Step 1: Find secondary obtuse angle $B_2$:     \nB_2 = 180^\circ - 61^\circ = 119^\circ\n
  • Step 2: Find secondary third angle $C_2$:     \nC_2 = 180^\circ - 61^\circ - 83^\circ = 36^\circ\n
  • Step 3: Solve for secondary side $c_2$ using the Law of Sines:     \n\frac{\sin(36^\circ)}{c_2} = \frac{\sin(119^\circ)}{30.6}\n
  • Rearrange equation:     \nc_2 = \frac{\sin(36^\circ) \times 30.6}{\sin(119^\circ)}\n
  • Computation result:     \nc_2 = 20.564 \approx 20.1\,\text{cm}\n
  • Verification: The calculated value $20.1\,\text{cm}$ is smaller than the original side length $35\,\text{cm}$, confirming that two valid triangles fit the given measurements.

Trigonometric Quadrants and Angle Ranges

• The standard Cartesian coordinate system divides angles into four distinct quadrants:
  • Quadrant I: $0^\circ \le \theta \le 90^\circ$
  • Quadrant II: $90^\circ < \theta \le 180^\circ$
  • Quadrant III: $180^\circ < \theta \le 270^\circ$
  • Quadrant IV: $270^\circ < \theta \le 360^\circ$ (or expressed as negative angle values)
• When evaluating trigonometric functions across quadrants, values must strictly fall within the designated bounds for the corresponding quadrant. An angle value outside $180^\circ$ to $270^\circ$ is invalid for Quadrant III.

Questions & Discussion

• Question: How is it known when to check if multiple triangles exist?

  • Response: Whenever presented with an $SSA$ (Side-Side-Angle) condition, checking for multiple triangles is required automatically.

• Question: Will students ever need to deal with three triangles?

  • Response: No, scenarios involving three triangles are not covered or required.

• Question: Is angle $C_1$ the whole angle of the entire triangle? What do $C_1$ and $C_2$ represent visually?

  • Response: Angle $C_1$ represents the angle at vertex $C$ when angle $B$ is acute ($B_1 = 61^\circ$), giving $C_1 = 94^\circ$. If the swinging arm moves inward such that angle $B$ becomes obtuse ($B_2 = 119^\circ$), angle $C$ shrinks to $C_2 = 36^\circ$

• Question: What does side $c_2$ solve for in the diagram?

  • Response: Side $c_2$ represents the smaller base length segment on the left when the swinging arm pivots inward to form the obtuse triangle.

• Question: Why is a given angle calculation considered wrong in the context of quadrants?

  • Response: For Quadrant III, valid angles must range strictly between $180^\circ$ and $270^\circ$. Any calculated value outside this boundary makes no mathematical sense for the third quadrant.

• Contextual & Practical Notes:

  • Reference sheets listing procedural steps in sequential order are recommended for students who struggle with mathematics.
  • Small classrooms can get significantly warmer compared to large lecture rooms.
  • Physical injuries can be patched up using electrical tape as an effective alternative to medical tape.
  • Optical prescriptions such as $0.25$ prescription glasses may be required when viewing small board writing from a distance.