78 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



Given Sought Formula 



1 , . i 9 sin ]8 sin 7 

 ^4 ^4 = - #6 sin 7 = - a 2 L - 



2 2 sin a 



a sin 

 a, b, 7 a tan a = 



a cos 7 



a, J(a + ft = 90 - iy. 



tan i(a - 0) = |^ cot J 



= (a 2 + b 2 - 2ab cos 7)*. 



= {(a + b) 2 - ^ab cos 2 



= {(a- b) 2 + 4ab sin 2 47}*. 



a b . -=- sin 



where tan <6 = 



cos 9 



_ g sin 7 

 sin a 



A = % ab sin 7. 



SOLUTION OF SPHERICAL TRIANGLES 



3.51 Right-angled spherical triangles. 



a, b, c = sides of triangle, c the side opposite 7, the right angle. 

 a, /3, 7 = angles opposite a, b, c, respectively. 



3.511 Napier's Rules: 



7T 



The five parts are a, b, co c, co a, co j8, where co c = - c. The right angle 



7 is omitted. 



The sine of the middle part is equal to the product of the tangents of the 

 adjacent parts. 



The sine of the middle part is equal to the product of the cosines of opposite 

 parts. 



From these rules the following equations follow: 

 sin a = sin c sin a, 

 tan a = tan c cos j3 = sin b tan a, 

 sin b = sin c sin /3, 

 tan b = tan c cos a = sin a tan j8, 

 cos a = cos a sin /3, 

 cos j8 = cos b sin a, 

 cos c = cot a: cot 3 = cos a cos 6. 



