trigonometric formulas from series
and the parity relations (http://planetmath.org/EvenoddFunction)
These produce straightforward many other important formulae, e.g.
The value , as well as the formulae expressing the periodicity of sine and cosine, cannot be directly obtained from the series (1) and (2) — in fact, one must define the number by using the function properties of the and its derivative series (http://planetmath.org/PowerSeries).
has on the interval exactly one root (http://planetmath.org/Equation). Actually, as sum of a power series, is continuous, and (see Leibniz’ estimate for alternating series (http://planetmath.org/LeibnizEstimateForAlternatingSeries)), whence there is at least one root. If there were more than one root, then the derivative
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:
Accordingly, we can define the number pi to be the least positive solution of the equation , multiplied by 2.
Thus we have and . Furthermore, by (5),
and by (4),
Consequently, the addition formulas (3) yield the periodicities (http://planetmath.org/PeriodicFunctions)
|Title||trigonometric formulas from series|
|Date of creation||2013-03-22 18:50:47|
|Last modified on||2013-03-22 18:50:47|
|Last modified by||pahio (2872)|
|Synonym||series definition of sine and cosine|