XX USEFUL FORMULAS. 



4. Formulas for Solution of Spherical Triangles. 



a. Right angled spherical triangles. 



a, b, c =: sides of triangle, c being the hypotenuse, 

 a, ^, y = angles opposite to a, />, c, respectively, 



7 = 9°°- 



sin rt = sin <r sin a, sin b =^ sm c sin /?, 



tan a = tan c cos /B, tan b = tan r cos a, 



= sin ^ tan a, = sin <? tan (3 ; 



cos a = cos a sin /S, cos /? = cos (^ sin a ; 



cos c = cos « cos b = cot a cot ft. 



b. Oblique angled triangles. 



a, b, c = sides of triangle, 



a, ^, y =r angles opposite to a, b, c, respectively, 

 ^ = i (« + ^ + ^), 



6=a-|-/3-(-y — 180° := spherical excess, 

 S ■=^ surface of triangle on sphere of radius r. 



sin a sin b sin c 

 sin a sin /S sin y 



cos n = cos Z-- cos r -\- sin /' sin c cos a, 



. „ , — cos (T cos (o- — a) „ , COS (or — 6) COS (o" — y) 



sin^ ir a = . — o • » cos^ h a:= ^ — ■ ^ ■ — ^ ^, 



2 sin ytJ sin y 2 sii^ ^ gjj^j ^ 



„ , — cos cr COS (o- — a) 

 tan^ ha — ^ ^ 



cos (o- — )8) COS (o- — y) 



. „ , sin (j — b) sin (j- — r) „ sin j sin (s — a) 



sin^ ho. = ^ — -. — J—. — ^ -, cos^ i a = = — T-^. -> 



^ sm b sm c ^ sin b sin c 



sin (s — b) sin (^ — c) 

 sin J- sm {s —a) 



cot i a cot i^ ^ + cos y 



cot i e = "^ — ■ — ■— ' -f 



^ sin y 



tan'^ 4 e = tan |- j tan \ {s — a) tan i (j — F) tan i (^ — c). 



100 



Napier's analogies. 



, / 17 cos h (a — 6) , . ,, sin J^i- (a — iS) 



tzul(a + ^) = ^-^^_p^^ tan ^ ., tan ^ (^ - ^) = 3-^ ^ („ _^ ^) tan J., 



, / I «N cos i- (« — /;) ^ „^ sin ^ (a — b) 



tan 1 (a + /S) = ^-) — j-,-( cot }, y, tan ?, (a — B) = -. — f ; , , ( cot i y. 



