Chapter3_Spherical_Trigonometry.pptx presentation

Chapter 3: Spherical Trigonometry

Historical background and significance

Unit sphere, spherical triangles, great circles

Laws of sines, cosines, and polar triangles

Applications in navigation and geodesy

Worked examples and exercises

1. Chapter 3: Spherical Trigonometry This chapter introduces the geometry of triangles drawn on a sphere. It is fundamental in navigation, astronomy, and geodesy for calculating distances and bearings on the Earth.

2. Historical Background Spherical trigonometry was developed by Islamic scholars like Ibn Muadh al-Jayyani (989–1079). Their work allowed accurate Earth size measurement and influenced European navigation.

3. Unit Sphere and Spherical Triangle A unit sphere has radius = 1. Spherical triangles are formed by intersecting great circles. Each side and angle is measured in degrees or radians. The sum of angles exceeds 180° by a 'spherical excess'.

4. Key Formulae Law of Sines: sinA/sina = sinB/sinb = sinC/sinc. Law of Cosines: cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A). Five-Part Formula: Relates sides and angles using trigonometric relationships.

5. Polar Triangle A polar triangle complements a spherical triangle by interchanging sides and angles (a ↔ 180°–A). This extends solving techniques and simplifies many navigation problems.

6. Radius of Earth and Nautical Mile Earth’s mean radius for spherical models: R = 6,367 km. 1 Nautical Mile = 1 arc minute of latitude = 1852 m. Conversions use radians: distance = angle (radians) × R.

7. Applications in Navigation Used to compute great-circle distances and bearings between two geographic points. Shortest path between two locations on a sphere is along a great circle.

8. Worked Example Example: Flight from Durban (29.53°S, 31.00°E) to Perth (31.58°S, 115.49°E). Great circle distance ≈ 7,858 km. Start bearing = 116°, arrival bearing = 66°. Shows bearing changes along great-circle routes.

9. Waypoints and Rhumb Lines Great-circle paths require constant bearing adjustments. Waypoints divide the route into smaller segments with fixed bearings (rhumb lines). These are plotted on Mercator projection charts.

10. Summary Spherical trigonometry links Earth geometry and navigation. Essential formulae: Sine, Cosine, Polar relationships. Applications: calculating distances, bearings, and waypoints for aircraft and ships.