Haversine Function is Even
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Theorem
The haversine is an even function:
- $\forall \theta \in \R: \map \hav {-\theta} = \hav \theta$
Proof
| \(\ds \map \hav {-\theta}\) | \(=\) | \(\ds \dfrac 1 2 \paren {1 - \map \cos {-\theta} }\) | Definition of Haversine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {1 - \cos \theta}\) | Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds \hav \theta\) | 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.