Analogue Formula for Spherical Law of Cosines/Corollary
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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:
\(\ds \sin A \cos b\) | \(=\) | \(\ds \cos B \sin C + \sin B \cos C \cos a\) | ||||||||||||
\(\ds \sin A \cos c\) | \(=\) | \(\ds \cos C \sin B + \sin C \cos B \cos a\) |
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 $\triangle A'B'C'$.
We have:
\(\ds \sin a' \cos B'\) | \(=\) | \(\ds \cos b' \sin c' - \sin b' \cos c' \cos A'\) | Analogue Formula for Spherical Law of Cosines | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {\pi - A} \, \map \cos {\pi - b}\) | \(=\) | \(\ds \map \cos {\pi - B} \, \map \sin {\pi - C} - \map \sin {\pi - B} \, \map \cos {\pi - C} \, \map \cos {\pi - a}\) | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {\pi - A} \paren {-\cos b}\) | \(=\) | \(\ds \paren {-\cos B} \, \map \sin {\pi - C} - \map \sin {\pi - B} \, \paren {-\cos C} \, \paren {-\cos a}\) | Cosine of Supplementary Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin A \, \paren {-\cos b}\) | \(=\) | \(\ds \paren {-\cos B} \sin C - \sin B \, \paren {-\cos C} \, \paren {-\cos a}\) | Sine of Supplementary Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin A \cos b\) | \(=\) | \(\ds \cos B \sin C + \sin B \cos C \cos a\) | simplifying |
$\blacksquare$
and:
\(\ds \sin a' \cos C'\) | \(=\) | \(\ds \cos c' \sin b' - \sin c' \cos b' \cos A'\) | Analogue Formula for Spherical Law of Cosines | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {\pi - A} \, \map \cos {\pi - c}\) | \(=\) | \(\ds \map \cos {\pi - C} \, \map \sin {\pi - B} - \map \sin {\pi - C} \, \map \cos {\pi - B} \, \map \cos {\pi - a}\) | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {\pi - A} \paren {-\cos c}\) | \(=\) | \(\ds \paren {-\cos C} \, \map \sin {\pi - B} - \map \sin {\pi - C} \, \paren {-\cos B} \, \paren {-\cos a}\) | Cosine of Supplementary Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin A \, \paren {-\cos c}\) | \(=\) | \(\ds \paren {-\cos C} \sin B - \sin C \, \paren {-\cos B} \, \paren {-\cos a}\) | Sine of Supplementary Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin A \cos c\) | \(=\) | \(\ds \cos C \sin B + \sin C \cos B \cos a\) | simplifying |
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $11$. Polar formulae.