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With similar arguments Euler proved the following identities involving the circular
functions sine and cosine:
x
3
x
5
x
7
sin
x
=
x
−
/
3!
+
/
5!
−
/
7!
...
x
2
x
4
x
6
...
Therefore by comparing the series for the exponential function
e
x
and of the circu-
lar functions, it is easy to obtain the famous identity of Euler connecting the three
numbers
cos
x
=
1
−
/
2!
+
/
4!
−
/
6!
π
,
i
,
e
:
Ta b l e 5 . 5
Euler's trigonometric identity
e
ix
=
cos
x
+
i
sin
x
.
If we put
x
=
π
, from cos
(
π
)=
−
1andsin
(
π
)=
0, we easily get the following
celebrated formula, called
Euler's identity
.
Ta b l e 5 . 6
Euler's identity
e
i
π
+
1
=
0
.
According to Napier's logarithms, Euler's constant
e
is called the
natural base of
logarithms
. The operation log
a
x
of a number
x
, with respect to any base
a
∈
R
,
>
a
1, yields the value such that
a
log
a
x
=
x
. From this definition, the main properties of
Table 5.7 easily follow.
Ta b l e 5 . 7
Basic properties of logarithms
log
a
xy
=
log
a
x
+
log
a
y
log
a
x
y
=
y
log
a
x
log
a
b
=
−
log
b
a
log
a
1
/
x
=
−
log
a
x
log
b
x
=
log
b
a
log
a
x
All the definitions of basic arithmetical operations, plus those of exponentiation,
and logarithm, with respect to real bases, easily extend to rational and real numbers.
We conclude by remarking that logarithms, and circular functions, had an enor-
mous impact on the technological and social development of the 17th century. In
