Analogue Formula for Spherical Law of Cosines/Proof 1
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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
| \(\ds \sin c \sin a \cos B\) | \(=\) | \(\ds \cos b - \cos c \cos a\) | Spherical Law of Cosines | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos b - \cos c \paren {\cos b \cos c + \sin b \sin c \cos A}\) | Spherical Law of Cosines | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos b \paren {1 - \cos^2 c} - \sin b \sin c \cos c \cos A\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin^2 c \cos b - \sin b \sin c \cos c \cos A\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin a \cos B\) | \(=\) | \(\ds \sin c \cos b - \sin b \cos c \cos A\) | simplifying |
| \(\ds \sin a \sin b \cos C\) | \(=\) | \(\ds \cos c - \cos a \cos b\) | Spherical Law of Cosines | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos c - \cos b \paren {\cos b \cos c + \sin b \sin c \cos A}\) | Spherical Law of Cosines | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos c \paren {1 - \cos^2 b} - \sin b \sin c \cos b \cos A\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin^2 b \cos c - \sin b \sin c \cos b \cos A\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin a \cos C\) | \(=\) | \(\ds \cos b \sin c - \sin c \cos b \cos A\) | simplifying |
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $7$. The analogue formula.