values of complex cosine

Since the complex cosine functionMathworldPlanetmathzcosz  has the prime periodPlanetmathPlanetmath 2π, the cosine attains all of its possible values in one of its period strips, for example in the period strip

{z-πRe(z)<π}. (1)

For finding out which values the cosine function can attain in a period strip, we solve the equation  cosz=w,  where w is any complex numberMathworldPlanetmathPlanetmath. Using Euler’s formula (


the equation may be written as

(eiz)2-2weiz+1= 0. (2)

This is a quadratic equation in eiz, whence we obtain the two roots (


The product of the roots is 1, and therefore the roots are distinct from zero for all values of w. If we set


the other root is the inverse number


The solution of the equation


is then obtained by taking the complex logarithm


and the other solution of (2) is


In the period strip (1), we have one solution z1 and one solution z2, both obtained with the value  n=0  (except z2 in the case  φ=-π  with  n=-1).  In (1), the points z1 and z2 are situated symmetrically with respect the origin.  In the cases  w=1  and  w=-1,  the equation (2) has double roots  z=0  and  z=-π,  respectively; then we may say that z1 and z2 coincide. Anyhow, we have the

Theorem. In every period strip, cosine attains any complex value at two points.

Example. The solution of the equation  cosz=2  is obtained from  eiz=2±3. In the period strip (1) we get


Since  2±3  are inverse numbers of each other, we have as result the purely imaginary numbersMathworldPlanetmathz=±iln(2+3).

From trigonometry, we know that the real zeros of cosine are the odd multiples of π2; from these points, ±π2 belong to the period strip (1). Thus ±π2 are the only points of (1) where the cosine vanishes. Therefore, according to the preceding theorem, the well-known points

(2n+1)π2(n= 0,±1,±2,)

are the only zeros of the cosine function on the whole complex plane.

The values of complex cosine function may be transferred to the complex sine function by means of the complement formula


One can think all points of the z-plane to bear the corresponding value of cosine, and then one can translate the plane in the direction of the real axis the distance π2; then the values of the sine have been placed to their correct . So one has transferred also the above properties of cosine to sine.


  • 1 Ernst Lindelöf: Johdatus funktioteoriaan. Second edition. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1936).
  • 2 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava, Helsinki (1963).
Title values of complex cosine
Canonical name ValuesOfComplexCosine
Date of creation 2013-03-22 17:36:29
Last modified on 2013-03-22 17:36:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Topic
Classification msc 33B10
Classification msc 30A99
Related topic RealPart
Related topic PropertiesOfQuadraticEquation
Related topic TakingSquareRootAlgebraically
Related topic ComplexLogarithm
Defines period strip