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Proof of sine formula

المؤلف:  A. Roy, D. Clarke

المصدر:  Astronomy - Principles and Practice 4th ed

الجزء والصفحة:  p 54

18-7-2020

2059

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)