analytical and geometrical Methods of Investigation. gg 
any preparatory process to shew that the value of the first co- 
efficient must = i.* 
VIII. Euler demonstrated this formula to be true, viz, 
T“=sin. arc — \ sin. 2 arc -J- •§■ sin. 3 arc — f sin. 4 arc + &c. 
The following is its analytical deduction, 
z ' =z { s w-ft ) + *■{ 1 } 
f 
= z-i 
i £t V ~' + 1 1 Z [ £-zV- t + J 
£zV~ % — ^ 2 i/JZ 7 -j - pzY—i . — &c. 1 
. + £~ % ^~ x — • j— 22^— 1 -|- £— 3st/“ — See.) 
y,% 
1 £32^—1 
— 4 - — &c. 
2 a /j 
1 
V— 1 
L 
— l 
+ L_ 
£-32 
y: 
■+ &c - J 
and — — . (2%/ — 1) *.| £ 2 v'_ 1 —£-21/— 1 1 — i (2 V 7 — i)“ s . 
{ }• + -§-(2 v / ^T) -I | £»*y-=r — £32-/rr J _ 
which is the analytical translation of Euler's formula. 
IX. Euler likewise shewed that 
sin. x = 2”. cos. cos. . cos. ~ . cos. ~L- • s i n> -JL, e 
Which may be thus demonstrated, 
sin. x = ( 2 v/ — 1 )— 1 / £ X V- 
£- 
■xV — t 
but (2 V 7 — 1 ) 1 j f *V-i £ x ^/ 1 | = 2 . 2“^ — i _|_ £ — sv/ — I j 
( 2 v~ }- x e y-r } 
• 2 ““ 1 f ~^~ 1 J . 2 . 2 — ) . 
y— « 
} 
_ * See Lagrange, Fonctions Analytiques. p. 2 6 , Lacroix, Traite du Calcul. dift 
ferentiel. Sec. p* 56. Le Seur, Sur le Calcul. diff. p. 105. Euler, Anal. Inf. 
Art. 153, 134. 
