tract 16. TRIGONOMETRY WITHOUT TABLES.

261

circumscribing the triangle: Then the value of A is equal to_ a a 2 3 a 5 3 ,5a 7 3.5.7a 9 „ „

' ' 2957 ' 95 x > + 244 + £Z54 + 2.4.6.7r 7 + 2.4.6.8.91*

For, since 2a is the chord of the arc on which the angle,whose measure is a, insists ; a will be the sine of half that arc,or the sine of the angle to the radius r, since an angle in thecircumference of a circle is measured by half the arc on whichit stands; now it is well known that the said half arc s isequal to

a+ H———H- 6 - &c : and, 3T4159r denoting

half the circumference of the same circle, or the arc of 180degrees, it will he

180z 51-2951795%

as 3-14159?” : 180° : : s

= SI-2951195 X(~ r + —- +r 2.3 r 2

degrees in the angle or half arc.

3-14159/’ — r

3 a> 3.5 a? „ , ,

4 iTSa? &c -> tlle

Corollary 1.—By reverting the above series, we obtain

a

r

A

n

: +

2.3n 2 2.3.4.5ft s

putting n = 51-2951195 =

2.3.4.5.6.7ft 7180

T41 59 &C.

See ;

Corollary 2.—If 2a he the hypothenuse of a right-angledtriangle, a will he = r, and then the general series will be-come n x (a + ‘

1

2 .? +

3.5 „ x 90 90x 3,14159 &c

-1-&c) — 90, or — =----

24.6.7 1 ’ ft LbO

2,4.5

3-14159 &c 1 3 3.5 2.5 7

2 t 2.3 + 2.4.5 f 24.6.7 ^ 2.4-.6.S.9

Carol. 3.—Since the chord of 60 degrees is = the radius, orthe sine of 30 degrees = half the radius, putting a for -*-r in the

general series, will give n x (- + —r~ 7 I FoTTTTs F T ~, ' ^ -

Stc 30; and hence the sum of the infinite series