Graphics Reference
In-Depth Information
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.