Spherical Law of Haversines
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Theorem
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
- $\hav a = \map \hav {b - c} + \sin b \sin c \hav A$
where $\hav$ denotes haversine.
Proof
\(\ds \cos a\) | \(=\) | \(\ds \cos b \cos c + \sin b \sin c \cos A\) | Spherical Law of Cosines | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 - 2 \hav a\) | \(=\) | \(\ds \cos b \cos c + \sin b \sin c \paren {1 - 2 \hav A}\) | Cosine in Terms of Haversine | ||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {b - c} - 2 \sin b \sin c \hav A\) | Cosine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 \map \hav {b - c} - 2 \sin b \sin c \hav A\) | Cosine in Terms of Haversine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \hav a\) | \(=\) | \(\ds \map \hav {b - c} + \sin b \sin c \hav A\) | simplifying |
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $13$. The haversine formula.