Plane trigonometry, by S.L. Loney.

456

TRIGONOMETRY.

that of the sine. The principle is therefore true except
when the angle is nearly zero, in which case the differences
are insensible and irregular.
383. Natural Tangents. With the same notation
as before we have
tan 0 + tan k tan k sec2 0
tan (O + k)- tan 0 = - tan 0 =
1-tan tank 1 - tan 0 tan k
= tan k sec20 (1 + tan 0 tan k + tan2 0 tan2 k...)
= sec2 + +... 1 +tan 0 k+3+ )
+ tan2 0(k2 +..)] (Art. 281)
==ksec2 si 0 + sec20 1 +tan20 +...... (1)
The third and higher terms may be omitted as before,
except when 0 is nearly a right angle.
sin 0
Unless the quantity k2 s- be large we shall then
have
tan ( + k)- tan 0 = k sec2 0............(2),
and the rule is approximately true.
When 0 is > 7 the second term of the equation (1) is
4
> 2k2, so that taking the greatest value of k, viz. about
~0003, this would give a significant figure in the seventh
place. The principle is therefore not true for angles
greater than -, when the differences of the tabulated
angles are 1'.