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.

~ 11. 12.

Algebraic and Transcendental.

21

3) If the size of the angle be extended by turning the leg further,
it follows that for any positive integer k:
sin (x + 2k)t) = sin x; cos (x + 2k)r) = cos x.
4) If the rotation be effected in an opposite sense, the angle is to
be denoted by a negative number; therefore we have then:
sin (- x) = sin(2 z - x) - sinx
cos (-x) = cos (2 - - x)= cos X,
and generally:
sin (x -- 27 k ) = sin x; cos (x - 2 k z) = cos x.
Both functions, which always lie between the values + 1 and - 1,
have the property of reproducing their values, whenever the independent variable is increased or diminished by an integer multiple of 2z.
Such a function is called periodic and 2z its period.
5) It is proved geometrically that:
sin (x ~- x) = sin x cos x1 - sin xl cos x
cos (x + xI) = cos x cos t sill sin ll X,
from which follow
sin -sinxl 2sin (x- x) cos (X + X,)

betv
/

cos x - cos x -- 2 sin j (x - x,) sin (x (- + ).
6) The area of the circular sector BCAM, whose angle lies
7een 0 and I X, is greater than the area of the triangle ABM and
less than the sum of the triangles AD M and
D7 BD3I.
Let the radius AM= 1, the angle AM IC= x,
__ - z ^ then A E =- sin x, JAIE = cos x; further:
AD: AM = ABE: JE,
a- so — I sin x
i -T and so AD= csx; accordingly for the areas
etioned e have the inequality
mentioned we have the inequality:

six 1 x
Fig. sin > X > sin x cos x or > cos x.
cos X Cos x sin x
This inequality holds, however small x is assumed; it holds also
if x be made negative. The quotient s is of such a nature,
sin x
that its numerator and denominator can become infinitely
small, while the quotient itself converges to a determinate
finite value; since for x = 0 cos x is = 1. Thus the value c
Cos X
which forms a superior limit, coincides with the lower limit cos x for
x = 0. Therefore also the included value will be 1, that is
Lim x 1 for ==0.
sin x