TRIGONOMETRY 



79 



3.52 Oblique-angled spherical triangles. 

 a, b, c = sides of triangle. 



a, ft, 7 = angles opposite to a, b, c, respectively. 

 s = \ (a + b + c), 

 a- = i (a + ft + 7), 



= Oi + /3-\-y 180=?= spherical excess, 

 S = surface of triangle on sphere of radius r. 



Given 

 a, b, c 



a, ft 7 



a, c, a 



Ambiguous case. 

 Two solutions 

 possible. 



a, 7, c 



Ambiguous case. 

 Two solutions 

 possible. 



Sought 

 a. 



sn 



a 



tan 2 - a = 

 2 



cos 2 - a = 

 2 



haversin a. = 



Formula 

 haversin a, 



sin (s b) sin (s c) 

 sin b sin c 



sin (s b) sin (s c) 

 sin 5 sin (s a) 



sin 5 sin (s a) 

 sin b sin c 



hav a hav (b c) 



sin b sin c 

 sin 2 J a = haversin a, 



cos cr cos (cr - a) 

 sin ft sin 7 



2 i - cos cr cos (cr - CE) 



Lctii CL == 7 >yv \" ' 



2 cos (cr p) cos (cr 7) 



cos (cr - ft) cos (cr - 7) 

 sin ft sin 7 



cos 2 - a 

 2 



sin 7 = 



sn a sn 

 sin a 



&< 



sn 



sin 



tan 6 = tan a cos c. 

 + 6) = sin tan c cot a 



cot 

 + < 



tan c cos a. 

 cos a sin <j) 



cos 



sin c = 



sin a sin 7 

 sin a. 



