How do you find the tangent of an imaginary number?

Answer 1

The answer is relatively simple if you know hyperbolic functions.

Suppose x is real so that ix is an imaginary number.

Then tanh x = -i*tan(ix)

So tan(ix) = (tanh x )/-i = i*tanh x

= i * sinh x/csh x = i*(ex - e-x)/(ex + e-x) = i*(1 - e-2x)/(1 + e-2x)