Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Geometry and Trigonometry | Topic 3.2
📋 Learning Intentions
- Apply the Sine Rule to find unknown sides and angles in non-right-angled triangles.
- Apply the Cosine Rule to find unknown sides and angles when two sides and the included angle, or all three sides, are known.
- Calculate the area of a triangle using the formula \( A = \frac{1}{2}ab\sin C \) when two sides and the included angle are given.
- Identify the ambiguous case of the Sine Rule and determine whether one or two valid triangles exist.
- Select the appropriate rule (Sine or Cosine) given the information provided in a problem.
- Solve multi-step problems in real-world contexts involving bearings, distances, and areas.
Real-World Introduction
Surveyors mapping a coastline, engineers designing a bridge span, and navigators plotting a ship's course all face the same mathematical challenge: finding distances and angles in triangles that contain no right angle. The standard SOHCAHTOA ratios apply only to right-angled triangles, so a more powerful set of tools is needed.
The Sine Rule and Cosine Rule extend trigonometry to any triangle. A GPS system, for example, uses trilateration — a process built on the same triangle geometry — to fix a position from three known distances. Architects calculating the roof pitch of an irregular polygon rely on these rules to decompose the shape into triangles and find every required measurement.
🌍 Why This Matters A search-and-rescue helicopter is dispatched from base. The pilot knows the bearing and distance to two landmarks but needs to find the direct distance between those landmarks to plan a fuel-efficient route. That calculation requires the Cosine Rule — no right angle exists in the triangle formed by the three locations.
Core Concepts & Definitions
Before applying any rule, it is essential to label a triangle consistently. The convention used throughout IB Mathematics AI is that side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\). This labelling is assumed in every formula in the booklet.
📐 Key Definition — Triangle Labelling Convention In triangle \(ABC\):
- Side \(a\) is the side opposite vertex \(A\) (i.e. side \(BC\)).
- Side \(b\) is the side opposite vertex \(B\) (i.e. side \(AC\)).
- Side \(c\) is the side opposite vertex \(C\) (i.e. side \(AB\)).
- Angles \(A\), \(B\), \(C\) are the interior angles at the respective vertices.
- The sum of interior angles: \( A + B + C = 180° \).
The Sine Rule
When a triangle has a known angle paired with its opposite side, a ratio is formed that is