Editing Complex number (section)

===Complex sine and cosine===
The series defining the real trigonometric functions [[sine|{{math|sin}}]] and [[cosine|{{math|cos}}]], as well as the [[hyperbolic functions]] {{math|sinh}} and {{math|cosh}}, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as [[tangent (function)|{{math|tan}}]], things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of [[analytic continuation]].

The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas.  For {{math|1=''z'' = ''x'' + ''iy''}},

<math display=block>\sin{z} = \sin{x} \cosh{y} + i \cos{x} \sinh{y}</math>

<math display=block>\cos{z} = \cos{x} \cosh{y} - i \sin{x} \sinh{y}</math>

<math display=block>\tan{z} = \frac{\tan{x} + i \tanh{y}}{1 - i \tan{x} \tanh{y}}</math>

<math display=block>\cot{z} = -\frac{1 + i \cot{x} \coth{y}}{\cot{x} -i \coth{y}}</math>

<math display=block>\sinh{z} = \sinh{x} \cos{y} + i \cosh{x} \sin{y}</math>

<math display=block>\cosh{z} = \cosh{x} \cos{y} + i \sinh{x} \sin{y}</math>

<math display=block>\tanh{z} = \frac{\tanh{x} + i \tan{y}}{1 + i \tanh{x} \tan{y}}</math>

<math display=block>\coth{z} = \frac{1 - i \coth{x} \cot{y}}{\coth{x} - i \cot{y}}</math>

Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct as [[Limit (mathematics)|limit]]s.