Sign of Haversine

Theorem

The haversine is non-negative for all $\theta \in \R$.

Proof

The haversine is conventionally defined on the real numbers only.

We have that:

$\forall \theta \in \R: -1 < \cos \theta < 1$

and so:

$\forall \theta \in \R: 0 < 1 - \cos \theta < 2$

from which the result follows by definition of haversine.

$\blacksquare$

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $13$. The haversine formula.