Angle of Spherical Triangle from Sides

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.