Napier's Analogies/Also presented as
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Napier's Analogies
Napier's Analogies can also be seen presented as:
| \(\text {(1)}: \quad\) | \(\ds \dfrac {\tan \frac {a + b} 2} {\tan \frac c 2}\) | \(=\) | \(\ds \dfrac {\cos \frac {A - B} 2} {\cos \frac {A + B} 2}\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \dfrac {\tan \frac {a - b} 2} {\tan \frac c 2}\) | \(=\) | \(\ds \dfrac {\sin \frac {A - B} 2} {\sin \frac {A + B} 2}\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds \dfrac {\tan \frac {A + B} 2} {\cot \frac C 2}\) | \(=\) | \(\ds \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2}\) | |||||||||||
| \(\text {(4)}: \quad\) | \(\ds \dfrac {\tan \frac {A - B} 2} {\cot \frac C 2}\) | \(=\) | \(\ds \dfrac {\sin \frac {a - b} 2} {\sin \frac {a + b} 2}\) |
It is supposed that they could also be presented as:
| \(\text {(1)}: \quad\) | \(\ds \tan \frac {a + b} 2 \cos \frac {A + B} 2\) | \(=\) | \(\ds \cos \frac {A - B} 2 \tan \frac c 2\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \tan \frac {a - b} 2 \sin \frac {A + B} 2\) | \(=\) | \(\ds \sin \frac {A - B} 2 \tan \frac c 2\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds \tan \frac {A + B} 2 \cos \frac {a + b} 2\) | \(=\) | \(\ds \cos \frac {a - b} 2 \cot \frac C 2\) | |||||||||||
| \(\text {(4)}: \quad\) | \(\ds \tan \frac {A - B} 2 \sin \frac {a + b} 2\) | \(=\) | \(\ds \sin \frac {a - b} 2 \cot \frac C 2\) |
but it has not been established that this appears anywhere in the literature.