Page:Mécanique céleste Vol 1.djvu/14

viii The following formulas are much used in the course of this work. They are to be found in most treatises on Trigonometry, and may be demonstrated by the method given in the appendix to this volume.

$Radius = $ 1. [1] Int.$$\sin.^2 z = -\frac{1}{2}\cdot \lbrace\cos. 2z - 1\rbrace$$,

[2] $$\sin.^3 z = -\frac{1}{2^2}\cdot\lbrace\sin. 3z - 3 \cdot\sin. z\rbrace$$,

[3] $$\sin.^4 z = \frac{1}{2^3}\cdot\lbrace\cos. 4z - 4 \cdot\cos. 2z + \frac{1}{2} \cdot\frac{4 \cdot3}{1\cdot2}\rbrace$$,

[4] $$\sin.^5 z = \frac{1}{2^4}\cdot\lbrace\sin. 5z - 5 \cdot\sin. 3z + \frac{5 \cdot 4}{1 \cdot 2} \cdot\sin. z \rbrace$$,

[5] $$\sin.^6 z = -\frac{1}{2^5}\cdot\lbrace\cos. 6z - 6 \cdot\cos. 4z + \frac{6 \cdot 5}{1 \cdot 2} \cdot\cos. 2z - \frac{1}{2} \cdot \frac{6 \cdot 5 \cdot 4}{1 \cdot 2 \cdot 3}\rbrace$$,

[6] $$ cos.^2 z = \frac{1}{2}\cdot\lbrace\cos. 2z + 1 \rbrace$$,

[7] $$ cos.^3 z = \frac{1}{2^2}\cdot\lbrace\cos. 3z + 3\cdot\cos. z \rbrace $$,

[8] $$ cos.^4 z = \frac{1}{2^3}\cdot\lbrace\cos 4z + 4\cdot\cos. 2z + \frac{1}{2}\cdot\frac{4\cdot3}{1\cdot2}\rbrace$$,

[9] $$ cos.^5 z = \frac{1}{2^4}\cdot\lbrace\cos. 5z + 5\cdot\cos. 3z + \frac{5\cdot4}{1\cdot2}\cdot\cos. z\rbrace$$,

[10] $$ cos.^6 z = \frac{1}{2^5}\cdot\lbrace\cos. 6z + 6\cdot\cos. 4z + \frac{6\cdot5}{1\cdot2}\cdot\cos.2z + \frac{1}{2}\cdot\frac{6\cdot5\cdot4}{1\cdot2\cdot3}\rbrace$$,$Hyp. log. c = $ 1

[11] $$ sin. z = \frac {c^{z\cdot\sqrt-1}-c^{-z\cdot\sqrt-1}}{2 \cdot \sqrt-1}$$,

[12] $$ cos. z = \frac {c^{z\cdot\sqrt-1}+c^{-z\cdot\sqrt-1}}{2}$$,

[13] $$ c^{z\cdot\sqrt-1} = \cos.z + \sqrt-1\cdot z + \sin.z $$, [14] $$c^{-z\cdot\sqrt-1}= \cos.z - \sqrt-1\cdot z + \sin.z $$,