262

FX.ANE TRIGONOMETRY

TRACT 16.

1

; +

; + ,

3.5

Ti &c >

is

2.3.2 3 2.4.5.2 s 2.4.6Z7.2 7

30 30x 3-14159 &c 3-14159 &c

n 180 6

■^th of the circumference of the circle whose diameter is 1.

Carol. 4.—It might easily be shown, from the principles ofcommon geometry, that the sine of 60 degrees is to the radius,as a-v/ 3 is' to 1; substituting then \rV3 for a in the general

1 3 3.3 2 3.5.3 s

series, we shall have n */3 x (—+-

2 ‘ 2.3.2 3 1 2.4.5.2 s + ’2.4.6.7.2 7&c) = 60; and hence the sum of the infinite series1 3 3 3 2 3.5.3 s

- +

, +

2 ‘ 2.3.2 3 2.4.5.2 s 1 2.4.6.7.2 7

60 60 X 3"14159 &c 3'14159 &c

See, will be

, and is therefore to

JJv 7 3 180v/3 3v/3

the infinite series in the 3d corollary, as 2 is to y 7 3.

Coral. 5 .—If b, c be the halves of the other two sides ofthe triangle, and b, c the degrees contained in their oppositeb H 1 3 b s

' 2,4.5r s

angles; since b — n x (-1—— + ■ " -■ &c), and c =

o . > ' v o c ) A. /ir 5 7

2.3 r 2

c C

n X (— + 3 &c, and the 3 angles of any triangle are equal

J 1 JitO r 5

to 180 degrees; we shall have 180r=A + B + c — n x,a\b\-c a i +b 1 +c 3

• +

r ' 2.3 r 3

a-\-b c I « 3 -fi 3 +c 3r ”*"2.3 5

&c), or the sum of the infinite series

' +

iz s + 6 s + c s 3.5 o'+ & + c”

2.4,5

+

2.4.6.7

„ , 180 180 x3-14159 &c

&c, will be =-=-——-- _ = 3.14159 &c = the

n 180

circumference of a circle whose diameter is 1 ; a, b, c, beingthe halves of the three sides of any triangle, and r the radiusof its circumscribing circle.

Coral. 6.—Since, bv theor. 3, b : a + c :: a — c : ——- =

7 - o

half the difference of the segments of the base (b) made by a'