Cosine of Half Side for Spherical Triangles
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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:
- $\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$.
Proof
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
- not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
- but also $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.
Let $s' = \dfrac {a' + b' + c'} 2$.
We have:
\(\ds \sin \dfrac {A'} 2\) | \(=\) | \(\ds \sqrt {\dfrac {\sin \paren {s' - b'} \sin \paren {s' - c'} } {\sin b' \sin c'} }\) | Sine of Half Angle for Spherical Triangles | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin \dfrac {\pi - a} 2\) | \(=\) | \(\ds \sqrt {\dfrac {\map \sin {s' - b'} \, \map \sin {s' - c'} } {\map \sin {\pi - B} \, \map \sin {\pi - C} } }\) | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {\dfrac \pi 2 - \dfrac a 2}\) | \(=\) | \(\ds \sqrt {\dfrac {\map \sin {s' - b'} \, \map \sin {s' - c'} } {\sin B \sin C} }\) | Sine of Supplementary Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos \dfrac a 2\) | \(=\) | \(\ds \sqrt {\dfrac {\map \sin {s' - b'} \, \map \sin {s' - c'} } {\sin B \sin C} }\) | Sine of Complement equals Cosine |
Then:
\(\ds s' - b'\) | \(=\) | \(\ds \dfrac {\paren {\pi - A} + \paren {\pi - B} + \paren {\pi - C} } 2 - \paren {\pi - B}\) | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi - \paren {A + B + C} } 2 + B\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac \pi 2 - \paren {S - B}\) | where $S = \dfrac {A + B + C} 2$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {s' - b'}\) | \(=\) | \(\ds \map \sin {\dfrac \pi 2 - \paren {S - B} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {S - B}\) | Sine of Complement equals Cosine |
and similarly:
- $\map \sin {s' - c'} = \map \cos {S - C}$
The result follows.
$\blacksquare$
Also see
- The other Half Side Formulas for Spherical Triangles:
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.100$