Plane trigonometry, by S.L. Loney.

RATIOS OF SMALL ANGLES.

263

230. From the tables it will be seen that the sine of
an angle and its circular measure agree to 7 places of
decimals so long as the angle is not greater than 18'.
They agree to the 5th place of decimals so long as the
angle is less than about 2~.
231. If 8 be the number of radians in an angle, which is less than a
03 02
right angle, then sin 0 is > 0 - 4 and cos 0 is > 1 - -
By Art. 227 we have
0 0
tan >. 0 0 0.006.sin > 2 cos 2.
Hence, since sin 0 = 2 sin 2 cos 2
we h.ave sin 0 > 0 Cos2, i.e >01- 2 \.
But since, by Art. 227,
But since, by Art. 227,. 0 0
sinm <
2 2'

therefore
Again,
therefore, since

1 - sin22>1- ) i.e. >1-4.
sin0>0 1 - ), i.e. >0- -.
0
cos 0=1 -.2 siDn2;
sin2 B <
2 2, 2-.
1- 2 sin2> 1-2 (), i.e. > 1 -.
2 ~_ I

we have

It will be found in a later chapter that
08 02 04
sin 0 > 0- -, and cos 0 < 1 - + 2
6 1 4-'

232. Ex. 1. Find the values of sin 10' and cos 10'.
Since '
Since 10'=6 - 180 x 6'