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series for arcsine from the term-by-term integration of the power series
for
(1
x
) and the used his repeated approximation method to
construct the series for sin
x
. This method is based on the formula
r
for the area of a circular sector and appears in Newton's notes from
1666. In
De Analysi
Newton's method for these series was based on a
calculation of arc length, which is closely related to the treatment in
this chapter. Again, it was Euler who recognised that an extension of
the trigonometric functions to complex variables required a formal
definition of these functions by series, and he obtained the equation
e
i sin
x
in 1748.
In 1821, Cauchy proved that a continuous solution of the
functional equation
f
(
x
y
)
cos
x
f
(
x
y
)
2
f
(
x
)
f
(
y
) must taketheform
cos
ax
or cosh
ax
with thecosineoccurring when
f
(
x
)
1 for some
x
.
In the late nineteenth ventury it was recognised that, if Weierstrass'
arithmetisation of analysis was to be carried through, a non-geometric
definition of these functions was required for real variables. In 1880
Thomae gave an analytic definition of cosine using Cauchy's equation
above. The series definition (as for complex variables) was also available
for this purpose, and the alternative definition with an integral (which
we have used) was commended by Felix Klein in 1908.
