Page:Mécanique céleste Vol 1.djvu/16

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INTRODUCTION.

[35 ] Int.

${\frac {\sin .a-\sin .b}{\sin .a+\sin .b}}={\frac {tang.{\frac {1}{2}}(a-b)}{tang.{\frac {1}{2}}(a+b)}},$

[36 ] "

${\frac {\sin .a-\sin .b}{\cos .a+\cos .b}}=tang.{\frac {1}{2}}\cdot (a-b),$

[37 ] "

${\frac {\sin .a+\sin .b}{\cos .a+\cos .b}}=tang.{\frac {1}{2}}\cdot (a+b),$

[38 ] "

${\frac {\sin .a+\sin .b}{\cos .a-\cos .b}}=-cot.{\frac {1}{2}}\cdot (a-b),$

[39 ] "

${\frac {\cos .a-\sin .b}{\cos .a+\cos .b}}=tang.{\frac {1}{2}}\cdot (a+b)\cdot tang.{\frac {1}{2}}\cdot (b-a),$

[40 ] "

${\frac {1-\cos .b}{1+\cos .b}}=tang.^{2}{\frac {1}{2}}b,$

[41 ] "

${\frac {\sin .a}{1+\cos .a}}=tang.{\frac {1}{2}}a,$

[41' ] "

${\frac {1+\cos .a}{\sin .a}}=cot.{\frac {1}{2}}a,$

[42 ] "

${\frac {\sin .a}{1+\cos .a}}=cot.{\frac {1}{2}}a,$

[43 ] "

$\sin .z=z-{\frac {z^{3}}{1.2.3}}+{\frac {z^{5}}{1.2.3.4.5}}-{\frac {z^{7}}{1.2.3.4.5.6.7}}+$ & $c.,$

[607d],

[44 ] "

$\sin .z=z-{\frac {z^{2}}{1.2}}+{\frac {z^{4}}{1.2.3.4}}-{\frac {z^{6}}{1.2.3.4.5.6}}+$ & $c.,$

[607e],

[45 ] "

$tang.z=z+{\frac {1}{3}}z^{3}+{\frac {2}{15}}z^{5}+{\frac {17}{315}}z^{7}+$ & $c.,$

[46 ] "

$z=arc.(\sin .y)=y+{\frac {1}{6}}y^{3}+{\frac {3}{40}}y^{5}+$&$c.,$

[47 ] "

$z=arc.(\cos .x)=(1-x)+{\frac {1}{6}}\cdot (1-x^{3})+{\frac {3}{40}}\cdot (1-x^{5})+$& $c.,$

[48 ] "

$z=arc.(tang.t)=t-{\frac {1}{3}}t^{3}+{\frac {1}{5}}t^{5}+$&$c.,$

[49 ] "

$dz=d\cdot (arc.\sin .y)={\frac {-dy}{\sqrt {1-yy}}},$

[50 ] "

$dz=d\cdot (arc.\cos .x)={\frac {dy}{\sqrt {1-xx}}},$

[51 ] "

$dz=d\cdot (arc.tang.t)={\frac {dt}{1+tt}},$