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trigonometric formulas from series∗
pahio†
2013-03-22 2:41:04
One may define the sine and the cosine functions for real (and complex)
arguments using the power series
sin x = x −
x3
x5
+
− +...,
3!
5!
(1)
cos x = 1 −
x4
x2
+
− +...,
2!
4!
(2)
and using only the properties of power series, easily derive most of the goniometric formulas, without any geometry. For example, one gets instantly from
(1) and (2) the values
sin 0 = 0,
cos 0 = 1
and the parity relations
sin(−x) = − sin x,
cos(−x) = cos x.
Using the Cauchy multiplication rule for series one can obtain the addition
formulas
(
sin(x+y) = sin x cos y + cos x sin y,
(3)
cos(x+y) = cos x cos y − sin x sin y.
These produce straightforward many other important formulae, e.g.
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x
(y =: x)
(4)
and
cos2 x + sin2 x = 1
(y =: −x).
(5)
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1
π
= 0, as well as the formulae expressing the periodicity of
2
sine and cosine, cannot be directly obtained from the series (1) and (2) — in
fact, one must define the number π by using the function properties of the cosine
series and its derivative series.
The equation
cos x = 0
The value cos
has on the interval (0, 2) exactly one root. Actually, as sum of a power series,
4
2
cos x is continuous, cos 0 = 1 > 0 and cos 2 < 1 − 22! + 24! < 0 (see Leibniz’
estimate for alternating series), whence there is at least one root. If there were
more than one root, then the derivative
− sin x = −x +
x3
x2
− + . . . = −x(1 −
+ − . . .)
3!
3!
would have at least one zero on the interval; this is impossible, since by Leibniz
the series in the parentheses does not change its sign on the interval:
1−
x2
22
+ −... > 1 −
> 0
3!
3!
Accordingly, we can define the number pi to be the least positive solution of the
equation cos x = 0, multiplied by 2.
Thus we have 0 < π < 4 and cos π2 = 0. Furthermore, by (5),
sin
π
= 1,
2
and by (4),
sin π = 0,
cos π = −1,
sin 2π = 0,
cos 2π = 1.
Consequently, the addition formulas (3) yield the periodicities
sin(x+2π) = sin x,
cos(x+2π) = cos x.
2