# trigonometric formulas from series

One may define the sine and the cosine functions for real (and complex) arguments using the power series

 $\displaystyle\sin{x}\;=\;x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-+\ldots,$ (1)
 $\displaystyle\cos{x}\;=\;1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-+\ldots,$ (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

 $\sin 0\;=\;0,\qquad\cos 0\;=\;1$

and the parity relations (http://planetmath.org/EvenoddFunction)

 $\sin(-x)\;=\;-\sin{x},\qquad\cos(-x)\;=\;\cos{x}.$

Using the Cauchy multiplication rule for series one can obtain the addition formulas

 $\displaystyle\begin{cases}\sin(x\!+\!y)\;=\;\sin{x}\cos{y}+\cos{x}\sin{y},\\ \cos(x\!+\!y)\;=\;\cos{x}\cos{y}-\sin{x}\sin{y}.\end{cases}$ (3)

These produce straightforward many other important formulae, e.g.

 $\displaystyle\sin 2x\;=\;2\sin{x}\cos{x},\qquad\cos 2x\;=\;\cos^{2}x-\sin^{2}x% \qquad(y\;=:\;x)$ (4)

and

 $\displaystyle\cos^{2}x+\sin^{2}x\;=\;1\qquad\qquad\qquad(y\;=:\;-x).$ (5)

The value  $\displaystyle\cos\frac{\pi}{2}=0$,  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 $\pi$ by using the function properties of the and its derivative series (http://planetmath.org/PowerSeries).

The equation

 $\cos{x}\;=\;0$

has on the interval$(0,\,2)$  exactly one root (http://planetmath.org/Equation).  Actually, as sum of a power series, $\cos{x}$ is continuous,  $\cos 0=1>0$  and  $\cos 2<1-\frac{2^{2}}{2!}+\frac{2^{4}}{4!}<0$  (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

 $-\sin{x}\;=\;-x+\frac{x^{3}}{3!}-+\ldots\;=\;-x(1-\frac{x^{2}}{3!}+-\ldots)$

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:

 $1-\frac{x^{2}}{3!}+-\ldots\;>\;1-\frac{2^{2}}{3!}\;>\;0$

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  $0<\pi<4$  and  $\cos\frac{\pi}{2}=0$.  Furthermore, by (5),

 $\sin\frac{\pi}{2}\;=\;1,$

and by (4),

 $\sin\pi\;=\;0,\qquad\cos\pi\;=\;-1,\qquad\sin 2\pi\;=\;0,\qquad\cos 2\pi\;=\;1.$

Consequently, the addition formulas (3) yield the periodicities (http://planetmath.org/PeriodicFunctions)

 $\sin(x\!+\!2\pi)\;=\;\sin{x},\qquad\cos(x\!+\!2\pi)\;=\;\cos{x}.$
Title trigonometric formulas from series TrigonometricFormulasFromSeries 2013-03-22 18:50:47 2013-03-22 18:50:47 pahio (2872) pahio (2872) 9 pahio (2872) Derivation msc 26A09 series definition of sine and cosine RigorousDefinitionOfTrigonometricFunctions ApplicationOfFundamentalTheoremOfIntegralCalculus TrigonometricFormulasFromDeMoivreIdentity GoniometricFormulae $\pi$