An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

22

Functionsa of a variabole.

Bk. I. ch. IV.

The quotient si xX travels through the series of the reciprocal
values and has likewise 1 for limit.
From these two functions two others are formed by division,
which are also immediately expressed as ratios of lengths by geometry:
sin x cos x
tan x-, cot x == i
cos x sin x 7
tanx is O for xz=0, +, + 2zr etc., cotx is 0 for x =-_ ~, - _-,
-+ 2 etc.; tan x increases beyond any assignable limit and becomes
+ oo in a determinate manner when x converges to the values + a- A,
+ - z, - rz etc., so does likewise cot x when x converges to the
values 0, ~+ z, ~ 2z etc. Both functions have the period r.
We have so far not given any convenient method of calculating
goniometric functions; only inasmuch as the determinateness of the
problem, to find for any given angle its sine etc., is known geometrically, we ventured to introduce the conception of these functions and
a symbolic notation for them.
13. From these four functions can be derived their inverses the
circular functions. If for any function we conceive the values of y
as given first, then the values of x appear as dependent; x is then
called the inverse function of y. In a table of logarithms we can
consider the logarithm as function of the number, but also inversely
the number as function of the logarithm.
The nature of a function may however be such that its inverse
function is defined only for isolated values of y. If we consider, for
instance, the function y = G(x), using G (x) to denote the greatest
integer in x, the inverse function is defined only for the integer
values =- 0, 1, 2,...; to each of these values then belong infinitely
many different values of x, namely to y = 0 all values from x = 0
to x = 1 exclusive, to y = 1 all values of x between 1 and 2, etc.
When y = sin x, the inverse denotes the angle x whose sine has
the given value y: it is written:
x == sin-ly
which is read, the angle whose sine is y, or the arc whose sine is y.
Similarly from the others x -= cos-ly, x =- tan-ly, x -- cot-ly.
Now since sine and cosine assume only values between - 1 and + 1,
sin-1 and cos-1 are defined only for the values of y within this
interval. Further, since different angles belong to the same value of
either sine, cosine, tangent or cotangent and so different numbers x,
circular functions are many-valued. In order to be able to consider
them as determinate numerical values in calculation, we adopt a convention which at the same time presents them as continuous functions:
Sin-'y denotes that number between - -1 and ~ z whose sine is y.
(- 1ly/ 1).