Applications (navigation, modelling) - Mathematics: Analysis & Approaches IB Study Notes

Navigation and modelling represent crucial real-world applications of trigonometry and geometry, forming a bridge between abstract mathematical concepts and practical problem-solving.

Bearings are angles measured clockwise from north, always expressed as three-digit numbers (e.g., 045°, 270°). They describe direction in navigation problems. The angle of elevation is measured upward from the horizontal to a point, while the angle of depression is measured downward from the horizontal.

Key Formulas for Navigation:

• Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (used when you have two angles and one side, or two sides and a non-included angle)
• Cosine Rule: $a^2 = b^2 + c^2 - 2bc\cos A$ (used when you have two sides and the included angle, or three sides)
• Area Formula: $\text{Area} = \frac{1}{2}ab\sin C$ (using two sides and the included angle)

Distance and Position Vectors: In 3D navigation, position vectors $\vec{r} = \begin{pmatrix} x \ y \ z \end{pmatrix}$ represent points in space. The distance between two points is $|\vec{AB}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.

Modelling assumptions are critical: we often assume Earth is flat for short distances, ignore air resistance, treat objects as particles, and assume constant speeds. Always state your assumptions explicitly in exam answers.

Mnemonic for bearing problems: "North Before Clockwise" — always measure from North, going clockwise.

These concepts enable solutions to problems involving ship navigation, aircraft flight paths, surveying, and architectural design.

Maritime Navigation: When a ship travels from port A on a bearing of 065° for 40 km, then changes course to bearing 140° for 30 km, we create a triangle where bearings become interior angles through careful angle conversion. The difference between consecutive bearings helps find the angle of the triangle. Think of bearings like a compass rose — each direction is precisely measured from true north.

Aviation Applications: An aircraft flying at 10,000 feet sees a runway at an angle of depression of 15°. Using tangent (opposite/adjacent), we model: $\tan 15° = \frac{10000}{d}$, giving horizontal distance $d = \frac{10000}{\tan 15°} \approx 37,321$ feet. This helps pilots calculate approach distances.

Surveying and Architecture: To find a building's height without measuring directly, stand at a known distance and measure the angle of elevation. If you're 50 m away and the angle is 32°, then $h = 50\tan 32° \approx 31.2$ m. This technique, called triangulation, was used to map entire countries.

Search and Rescue: When two coast guard stations simultaneously detect a distress signal, they can use bearings and the sine rule to locate the vessel. Station A measures bearing 042°, Station B (20 km east) measures bearing 315°. The triangle formed has calculable angles, revealing the exact position.

Real-world complexity: GPS systems use similar principles but in 3D with satellites. Your phone calculates your position by measuring distances to multiple satellites using the speed of light and time delays — essentially solving simultaneous equations with spheres instead of triangles.

Analogy: Navigation problems are like treasure maps where X marks the spot — you follow directions (bearings) for certain distances (magnitudes) to reach your destination.