Angle of Spherical Triangle from Sides
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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:
- $\cos A = \cosec b \cosec c \paren {\cos a - \cos b \cos c}$
Proof
\(\ds \cos b \cos c + \sin b \sin c \cos A\) | \(=\) | \(\ds \cos a\) | Spherical Law of Cosines | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin b \sin c \cos A\) | \(=\) | \(\ds \cos a - \cos b \cos c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos A\) | \(=\) | \(\ds \dfrac {\cos a - \cos b \cos c} {\sin b \sin c}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosec b \cosec c \paren {\cos a - \cos b \cos c}\) | Definition of Real Cosecant Function |
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $5$. The cosine-formula.