Cosine in Terms of Haversine
Jump to navigation
Jump to search
Theorem
- $\cos \theta = 1 - 2 \hav \theta$
where $\cos$ denotes cosine and $\hav$ denotes haversine.
Proof
\(\ds \hav \theta\) | \(=\) | \(\ds \dfrac 1 2 \paren {1 - \cos \theta}\) | Definition of Haversine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \hav \theta\) | \(=\) | \(\ds 1 - \cos \theta\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos \theta\) | \(=\) | \(\ds 1 - 2 \hav \theta\) |
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $13$. The haversine formula.