56.3: The Cosine Formula

The cosine formula from spherical trigonometry gives the angular separation between two points on the surface of a sphere, where the apex of the angle is at the center of the sphere. If the two points have latitudes $\phi_{1}$ and $\phi_{2}$ and longitudes $L_{1}$ and $L_{2}$, then the cosine formula gives the angular separation of the two points $\theta$ :

$$\cos \theta=\sin \phi_{1} \sin \phi_{2}+\cos \phi_{1} \cos \phi_{2} \cos \left(L_{1}-L_{2}\right)$$

This formula may be used to compute, for example, the angular separation between two stars in the sky, where $\phi=\delta$ and $L=\alpha$ are the celestial counterparts of latitude and longitude, called declination and right ascension, respectively (see Chapter 54). To find the distance $s$ between two points on the Earth's surface, convert $\theta$ to radians and use $s=R_{\oplus} \theta$, where $R_{\oplus}=6371.0 \mathrm{~km}$ is the average radius of the Earth.

If the angular separation $\theta$ between the two points is small, better accuracy may be obtained by using the haversine function, $\operatorname{hav}(x)$. The haversine is defined by

$$\operatorname{hav} \theta \equiv \sin ^{2}\left(\frac{\theta}{2}\right)$$

and so the inverse haversine function is given by

$$\operatorname{hav}^{-1} y \equiv 2 \sin ^{-1} \sqrt{y} \text {. }$$

Using the haversine function, the cosine formula can be replaced by

$$\operatorname{hav} \theta=\operatorname{hav}\left(\phi_{1}-\phi_{2}\right)+\cos \phi_{1} \cos \phi_{2} \operatorname{hav}\left(L_{1}-L_{2}\right)$$