# complex sine and cosine

We define for all complex values of $z$:

• $\displaystyle\sin{z}\;:=\;z\!-\!\frac{z^{3}}{3!}\!+\!\frac{z^{5}}{5!}\!-\!% \frac{z^{7}}{7!}\!+-\ldots$

• $\displaystyle\cos{z}\;:=\;1\!-\!\frac{z^{2}}{2!}\!+\!\frac{z^{4}}{4!}\!-\!% \frac{z^{6}}{6!}\!+-\ldots$

Because these series converge for all real values of $z$, their radii of convergence are $\infty$, and therefore they converge for all complex values of $z$ (by a known of Abel; cf. the entry power series), too.  Thus they define holomorphic functions in the whole complex plane, i.e. entire functions (to be more precise, entire transcendental functions).  The series also show that sine is an odd function and cosine an even function.

Expanding the complex exponential functions $e^{iz}$ and $e^{-iz}$ to power series and separating the of even and odd degrees gives the generalized Euler’s formulas

 $e^{iz}\;=\;\cos{z}+i\sin{z},\quad e^{-iz}\;=\;\cos{z}-i\sin{z}.$

Adding, subtracting and multiplying these two formulae give respectively the two Euler’s formulae

 $\displaystyle\cos{z}\;=\;\frac{e^{iz}\!+\!e^{-iz}}{2},\quad\sin{z}\;=\;\frac{e% ^{iz}\!-\!e^{-iz}}{2i}$ (1)

(which sometimes are used to define cosine and sine) and the “fundamental formula of trigonometry

 $\cos^{2}{z}+\sin^{2}{z}\;=\;1.$

As consequences of the generalized Euler’s formulae one gets easily the addition formulae of sine and cosine:

 $\sin{(z_{1}\!+\!z_{2})}\;=\;\sin{z_{1}}\cos{z_{2}}+\cos{z_{1}}\sin{z_{2}},$
 $\cos{(z_{1}\!+\!z_{2})}\;=\;\cos{z_{1}}\cos{z_{2}}-\sin{z_{1}}\sin{z_{2}};$

so they are in $\mathbb{C}$ fully as in $\mathbb{R}$.  It means that all goniometric formulae derived from these, such as

 $\sin{2z}\;=\;2\sin{z}\cos{z},\quad\sin{(\pi\!-\!z)}\;=\;\sin{z},\quad\sin^{2}{% z}\;=\;\frac{1-\cos{2z}}{2},$

have the old shape.  See also the persistence of analytic relations.

The addition formulae may be written also as

 $\sin{(x\!+\!iy)}\;=\;\sin{x}\cosh{y}+i\cos{x}\sinh{y},$
 $\cos{(x\!+\!iy)}\;=\;\cos{x}\cosh{y}-i\sin{x}\sinh{y}$

which imply, when assumed that  $x,\,y\in\mathbb{R}$,  the results

 $\mbox{Re}(\sin(x\!+\!iy))\;=\;\sin{x}\cosh{y},\quad\mbox{Im}(\sin(x\!+\!iy))\;% =\;\cos{x}\sinh{y},$
 $\mbox{Re}(\cos(x\!+\!iy))\;=\;\cos{x}\cosh{y},\quad\mbox{Im}(\cos(x\!+\!iy))\;% =\;-\sin{x}\sinh{y}.$

Thus we get the modulus estimation

 $\begin{array}[]{l}|\sin(x\!+\!iy)|\;=\;\sqrt{\sin^{2}{x}\cosh^{2}{y}+\cos^{2}{% x}\sinh^{2}{y}}\;=\;\sqrt{\sin^{2}{x}\cosh^{2}{y}+(1-\sin^{2}{x})\sinh^{2}{y}}% \\ \;=\;\sqrt{\sin^{2}{x}(\cosh^{2}{y}-\sinh^{2}{y})+\sinh^{2}{y}}\;=\;\sqrt{\sin% ^{2}{x}\,\cdot 1+\sinh^{2}{y}}\;\geq\;|\sinh{y}|,\end{array}$

which tends to infinity when  $z=x\!+\!iy$  moves to infinity along any line non-parallel to the real axis.  The modulus of $\cos(x\!+\!iy)$ behaves similarly.

Another important consequence of the addition formulae is that the functions $\sin$ and $\cos$ are periodic and have $2\pi$ as their prime period (http://planetmath.org/ComplexExponentialFunction):

 $\sin{(z\!+\!2\pi)}\;=\;\sin{z},\quad\cos{(z\!+\!2\pi)}\;=\;\cos{z}\quad\forall z$

The periodicity of the functions causes that their inverse functions, the complex cyclometric functions, are infinitely multivalued; they can be expressed via the complex logarithm and square root (see general power) as

 $\arcsin{z}\;=\;\frac{1}{i}\log(iz\!+\!\sqrt{1\!-\!z^{2}}),\quad\arccos{z}\;=\;% \frac{1}{i}\log(z\!+\!i\sqrt{1\!-\!z^{2}}).$

The derivatives of sine function and cosine function are obtained either from the series forms or from (1):

 $\frac{d}{dz}\sin{z}\;=\;\cos{z},\quad\frac{d}{dz}\cos{z}\;=\;-\sin{z}$

Cf. the higher derivatives (http://planetmath.org/HigherOrderDerivativesOfSineAndCosine).

 Title complex sine and cosine Canonical name ComplexSineAndCosine Date of creation 2013-03-22 14:45:25 Last modified on 2013-03-22 14:45:25 Owner pahio (2872) Last modified by pahio (2872) Numerical id 31 Author pahio (2872) Entry type Definition Classification msc 30D10 Classification msc 30B10 Classification msc 30A99 Classification msc 33B10 Related topic EulerRelation Related topic CyclometricFunctions Related topic ExampleOfTaylorPolynomialsForSinX Related topic ComplexExponentialFunction Related topic DefinitionsInTrigonometry Related topic PersistenceOfAnalyticRelations Related topic CosineAtMultiplesOfStraightAngle Related topic HeavisideFormula Related topic SomeValuesCharacterisingI Related topic UniquenessOfFouri Defines complex sine Defines complex cosine Defines sine Defines cosine Defines goniometric formula