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cot x
cos x
B
tan x
1
x
sin x
O
A
Figure11.1
ignore the geometric origins of the functions and define sine and cosine
by their power series: this is the procedure adopted by Burkill. It is also
possible to define the sine function by an infinite product or to develop
thecircular functions from thedefinition
dt
1 t
arctan x
:
this is the procedure adopted by Hardy.
But if the geometric origins of these functions are to be respected
we must develop a formal definition of angle either from the notion of
the area of a sector of a unit circle (the procedure adopted by Spivak)
or from a formal definition of arc length. This is what is done in qns
39 48. You may, if you wish, skip to thedefinition of circular arc
length following qn 48 and explore the intervening problems when your
curiosity is aroused.
Length of a line segment
39 Give an algebraic formula for the non-negative function
f :[ 1, 1] R whose graph will appear as a semicircle with centre
at theorigin and radius 1.
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