Four-Parts Formula
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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.
We have:
- $\cos a \cos C = \sin a \cot b - \sin C \cot B$
That is:
- $\map \cos {\text {inner side} } \cdot \map \cos {\text {inner angle} } = \map \sin {\text {inner side} } \cdot \map \cot {\text {other side} } - \map \sin {\text {inner angle} } \cdot \map \cot {\text {other angle} }$
This is known as the four-parts formula, as it defines the relationship between each of four consecutive parts of $\triangle ABC$.
Corollary
- $\cos A \cos c = \sin A \cot B - \sin c \cot b$
Proof
\(\ds \cos b\) | \(=\) | \(\ds \cos a \cos c + \sin a \sin c \cos B\) | Spherical Law of Cosines | |||||||||||
\(\ds \cos c\) | \(=\) | \(\ds \cos b \cos a + \sin b \sin a \cos C\) | Spherical Law of Cosines | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos b\) | \(=\) | \(\ds \cos a \paren {\cos b \cos a + \sin b \sin a \cos C} + \sin a \sin c \cos B\) | substituting for $\cos c$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos b\) | \(=\) | \(\ds \cos^2 a \cos b + \cos a \sin b \sin a \cos C + \sin a \sin c \cos B\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos b \paren {1 - \cos^2 a}\) | \(=\) | \(\ds \cos a \sin b \sin a \cos C + \sin a \sin c \cos B\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos b \sin^2 a\) | \(=\) | \(\ds \cos a \sin b \sin a \cos C + \sin a \sin c \cos B\) | Sum of Squares of Sine and Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cot b \sin a\) | \(=\) | \(\ds \cos a \cos C + \dfrac {\sin c} {\sin b} \cos B\) | dividing both sides by $\sin a \sin b$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cot b \sin a\) | \(=\) | \(\ds \cos a \cos C + \dfrac {\sin C} {\sin B} \cos B\) | Spherical Law of Sines | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos a \cos C\) | \(=\) | \(\ds \sin a \cot b - \sin C \cot B\) | simplification |
$\blacksquare$
Also see
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $8$. The four-parts formula.