NavList:

Hi,Hanno

Your Russian manual formula for calculating a computed altitude from celestial observation:
"sin(Hc) = sin(D)*cos(L) + cos(D)*sin(L)*cos(t)" is not correct- The right formula is:
sin(Hc)=sin(D)Sin(L)+cos(D)cos(L)cos(t).................(1)

( See,for instance, Dutton's "Nautical Navigation",15th ed.,page,302; or "2013 Nautical Almanac",page 270,and many others)
Now, let us use some plane trigonometry formulae:
cos(D-L)=cos(D)cos(L)+sin(D)sin(L)
-cos(D+L)=-cos(D)cos(L)+sin(D)sin(L)
____________________________________
By addition of both eqns .and rearrangement, we get,
[cos(D-L)-cos(D+L)]/2 = sin(D)sin(L).................(2)
Also,
cos(S-2D)+cos(S-2L)=2cos[(S-2D+S-2L)/2]cos[(S-2D-S+2L)/2]=
2cos[S-(D+L)]cos(D-L)= 2cos(t)cos(D-L) (Since S-(D+L)=t)...............(3)
Analogously,
cos(S-2t)+cos(S)=2cos[(S-2t+S)/2]cos[{S-2t-S)/2]
=2cos(S-t)cos(t)=2cos(D+L)cos(t) (since S-t=D+L)...............(4)
Let us add together eqns(3) and (4):
cos(S-2D)+cos(S-2L)+cos(S-2t)+cos(S)=4cos(t)cos(D)cos(L).
Whence, [cos(S-2D)+cos(S-2L)+cos(S-2t)+cos(S)]/4=cos(t)cos(D)cos(L)..............(5)
Finally, by adding up eqns (2) and (5) we get
[cos(D-L)-cos(D+L)]/2+[cos(S-2D)+cos(S-2L)+cos(S-2t)+cos(S]/4
=sin(D)sin(L)+cos(D)cos(L)cos(t)=sin(Hc).................(6)

Best regards,

Antonio Sauci.

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