OF PLANE TRIGONOMETRY, 
243 
hlSWyi 
pressed by a 
a a 
rr-3 + 2T3T 
&c. and its co-sine by 1 
4 . 5 
+ 
4.5.6.7 
2.3.4 
2 . 3 
+ 
&C. 
5.6 2. 3.4.5.6.7.8 
Thus, tor example, let it be required to find the sine 
of one minute: then, as losoo (the minutes in iso de 
grees). : 1 :: 3,14159265358 &c. : .000290888203665 
— file length of an arch ©f one minute: therefore, in 
this case, a — .000290888208665, and 
2 . 3 
<=-s-> 
— .000000000004102, &c. And consequently 
.000290888204563 zr the required sine of one minute. 
Again, let it be required to find the sine and co-sine 
of five degrees, each true to seven places of decimals. 
Here ,0002908882, the length of an arch of l minute 
(found above) being multiplied by 300, the number of 
minutes in 5 degrees, the product .08726646 will be the 
length ot an arch of 5 degrees: therefore, in this case, 
we have 
a — ,08726646 
a 3 
— — ,00011076, 
T 
= + ,00000004, 
&c. and consequently ,08715574 = the sine of 5 de 
crees. Also 
— =r ,00380771, 
a 4 
■ = ,00000241 ; 
24 
and consequently ,9961947 — the co-sine of 5 degrees. 
After the same manner, the sine and co-sine of any 
other arch may be derived ; but the greater the arch is 
the slower the series will converge, and therefore a 
greater number of terms must be taken to bring out the 
conclusion to the same degree of exactness, 
r 2