Zadanie 2f, Paper 3, maj 2025, TZ2, Matematyka AA IB HL

The method defines the spherical angle at W as the angle between the normals $\mathbf w\times\mathbf p$ and $\mathbf w\times\mathbf m$.

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west along the equator from N. An equatorial reference point E has vector $\mathbf e=(5,0,0)$. Station M lies north of N on the same meridian, with $\mathbf m=(0,5\cos\theta,5\sin\theta)$ and 0 < θ < π/2. The shorter surface distance NM is 3500 km. Station W is on the equator west of N, with $\mathbf w=(5\sin105^\circ,5\cos105^\circ,0)$. Surface distances follow the shorter great-circle arc.

The initial bearing from W towards M is the angle at W in spherical triangle WMP. You may use $\mathbf w\times\mathbf p=(25\cos105^\circ,-25\sin105^\circ,0)$. Use the method from (c) to find this bearing.