Proof of sine formula
المؤلف:
A. Roy, D. Clarke
المصدر:
Astronomy - Principles and Practice 4th ed
الجزء والصفحة:
p 54
18-7-2020
1756
Proof of sine formula
Rewrite the cosine formula (1) as
cos a = cos b cos c + sin b sin c cos A........ (1)
sin b sin c cos A = cos a − cos b cos c.
Squaring, we obtain
Now
Hence,
(2)
But the right-hand side of equation (2) is a symmetrical function of a, b and c so that we can write:
Dividing throughout by sin2 a sin2 b sin2 c we obtain
Taking the square root and remembering that in a spherical triangle all sides and angles are less than 180◦ in value, we obtain
.......(3)
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