The latitude and longitude of a point $P$ in the Northem Hemisphere are related to spherical coordinates $\rho, \theta, \phi$ as follows. We take the origin to be the center of the earth and the positive z-axis to pass through the North Pole. The positive $x$ -axis passes through the point where the prime meridian (the meridian through Greenwich. England) intersects the equator. Then the latitude of $P$ is $\alpha=90^{\circ}-\phi^{\circ}$ and the longitude is $\beta=360^{\circ}-\theta^{\circ}$ Find the great-circle distance from Los Angeles (lat. $34.06^{\circ} \mathrm{N}$. long. $118.25^{\circ} \mathrm{W}$ ) to Montréal (lat. $\left.45.50^{\circ} \mathrm{N}, \text { long. } 73.60^{\circ} \mathrm{W}\right) .$ Take the radius of the earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)
