Plane trigonometry, by S.L. Loney.

INDETERMINATE EXPRESSIONS.

335

Now, the smaller 0 is, the more nearly do both
1 sin 0
and
cos 0 0
approach to unity. Hence, when 0 is actually zero, the
given expression = 1 x 1 =1.
Such an expression as the one we have discussed is
said to be indeterminate. We should more properly say
that the expression is " at first sight" indeterminate.
286. In many cases the real value is very easily
found by using the series for sin 0 and cos 0. The method
is shewn in the following examples, of the first of which
the example in the preceding article is a particular case.
Ex. 1. Find the value of
n sin 0 - sin nO
0 (cos 0 - cos nO)'
The expression
03 _5 3 n5 3 05
- - -- + hig
n3 - n 5 5- n
03 - - 5 5- higher powers of 0
Q 02 + n 4higher powers of
n3 - n n5 - n02 + higher powers
Wh a is er2 +higher powers
When 0 is zero, this expression
n3 - n n2 - 1 n
13~ 2 =3'